Contemporary Mathematics
The Cohomology of the Mod 2 Steenrod Algebra: A
Computer Calculation
Robert R. Bruner
Abstract. The cohomology of the mod 2 Steenrod algebra for homologicaldegree s < 40 and internal degree t < 141 has been calculated by the techniqueof minimal resolutions. The most extensive previous calculation is that ofTangora, and is contained in pp. 18-56. In addition, we have calculated chainmaps induced by all the indecomposable elements in this range, and haveused this to determine the complete product structure. Even in the range inwhich the vector space structure was known due to Tangora’s work, some ofthe products, and more significantly, the knowledge that we now have all theproducts, is new. As we shall show in applications to the homotopy groups ofspheres, this product structure has many consequences.
1. Introduction
While the algebraic topologist who who simply wishes to determine the valueof a homotopy group may not care about the details of the programming, the datacan be trusted only if the algorithms and the programs implementing them arecorrect. Accordingly we will describe the main features of the programs which haveproduced these calculations. First, however, we describe how to read the tables.
2. Key to the Tables
The attached tables describe the cohomology of the Steenrod algebra in therange s < 40, t < 141, with all products. This has been computed using the pro-grams described in [1, 2]. The product structure has been calculated by computingchain maps for all the indecomposable elements found. The periodicity operatorP (x) =< h4, h
30, x > was computed from the chain map induced by the indecom-
posable x, when x is indecomposable, and from a null-homotopy of h30h4 in other
cases. The programs described in [1, 2] compute resolutions, that is Tor ratherthan Ext. Dualization, to produce Ext, and computation and sorting of productswas done using MAGMA, which greatly speeded the processing of an overwhelm-ing amount of data and eliminated the errors which would otherwise certainly havecrept in.
In the main chart, the product structure is complete, in that any product notlisted is zero. A filter to remove redundant products would shorten the entries atthe price of requiring multiple cross-references to check relations. Instead, we have
c©0000 (copyright holder)
1
2 ROBERT R. BRUNER
simply listed all nonzero monomials, sorted by their value in terms of the basisfound by the machine. Thus, the entry in the 37 stem
7 (10) h1t = h22n = c1f0
(01) h20x
(11) h3r
means that h3r = h1t + h20x. Within each entry, the monomials are listed in
lexicographic order, where the generators are ordered firstly by increasing filtration(cohomological degree), then by internal degree, then arbitrarily. This is the orderin which the generators are listed in the table of geneators. There, we give the namesof the 464 generators which appear in this range, their filtration and total degree(‘stem’), and their value in terms of the basis found by the machine. For example,in filtration 4, stem 18, we have the element f0, with value (10). This means thatExt4,22 is two dimensional and that f0 is the first basis vector in the machine’sinternal representation. By contrast, f1 has value (11). We made the latter choicebecause we are able to compute Sq0 in this case, and want f1 = Sq0(f0). In general,we have tried to choose generators in Sq0-families. Unfortunately, it is a difficultproblem to compute Sq0 mechanically at present. However, by using the connectionwith the root invariant we are able to identify a number of Sq0’s [3]. In particular,we are certain that ai+1 = Sq0(ai) for a ∈ {h, c, d, e, f, g, p, n, x, r, t, m, }, except form1 and t1, for which indeterminacy in the root invariant leaves us with two possiblechoices for each. Otherwise unambiguous identification of the first member of thefamily results in unambiguous identification of the whole family. Further, manyfamilies start in bidegrees which are one dimensional, so their definition is clear.The irritating exception is f0. The traditional definitions of f0 in terms of Masseyproducts all have nonzero indeterminacy, but there is an unambiguous definition interms of the Steenrod operations: f0 = Sq1(c0). Although computation of Steenrodoperations by mechanical means is out of reach in general, it is possible that wecould accomplish this one since it is in such a low dimension. We also suspect thatSq0(D3) = D31, Sq0(D1) = D11 and Sq0(H1) = H11, but cannot prove these bycurrently available means, so have resisted naming the elements as if this were so.There is the further difficulty that past naming conventions are rather incoherentin stems above 50, so that we would have clashes. For example D1, D2, and D3 arein filtrations 5,6, and 4, respectively. Clearly it is time to resort to more systematicnaming conventions. Finally, in the absence of any way to identify generatorswith elements already known in the cohomology of the Steenrod algebra, we haveresorted to precise but ugly names like x12,50, which means the 50th generator themachine found in filtration 12.
3. What Has Been Calculated
We have calculated a minimal resolution in the category of modules over theSteerod algebra for the cohomology of the sphere S0, the stunted projective spacesP−n for −55 < −n < 0, and Pn for n = 1, 2, 3, ???, and for the cofibers C2, Cη, Cν, ???.
In addition we have computed the chain maps induced by the inclusion ofthe bottom cell S−n −→ P−n and the −1-cell S−1 −→ P−n, as well as all 465indecomposable elements of ExtA(F2, F2). That is, for a set of multiplicative gen-
erators x ∈ Exts,tA (F2, F2) we have comupted the chain map induced by the cocycleCs −→ ΣtF2.
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 3
What is found in the tables and charts.
4. Algorithms and Data Structures
The resolutions and chain maps are organized in directories reflecting the cate-gorical relationships between them. That is, each module has a directory, containingits definition (dimF2
, degrees of the generators, the action of A on the generators).The programs conmstruct the minimal resolution of the module in this directory.Any chain maps to this resolution will produce temporary files in this directorycontaining cycles whose lifts are needed.
Each chain map has a subdirectory of its domain’s directory which containsthe definition of the chain map (its homological and internal degrees, it’s domainand codomain, and the cocycle). The program constructs the chain map in thisdirectory.
Similarly, a null homotopy of fg lives in a subdirectory of the map g, and pointsto f .
The processing is divided into two main stages. One stage is the calculationof the resolutions and chain maps. This is done by a small package of C programsdesigned to handle the rather large data structures which arise as efficiently aspossible. The other stage is the processing of this data to compute the groupsExtA(M, F2) and the homomorphisms ExtA(M, F2) −→ ExtA(N, F2). This is doneby shell scripts which do simple text manipulation and by MAGMA programswhich process the output of the shell scripts. The ability to write these programsin terms of data structures such as vector spaces and homomorphisms makes thesepprograms shorter and clearer, but generally less efficient. It is clear that the cal-cuations will gradually be rewritten in these higher level languages as the growthof computers compensates for the overhead associated with languages with sophis-ticated data types.
Data files defined by the user.Data files generated by the programs.(permanent and workfiles)Format of elements and files.
Algorithmsaddgengenimkergenimgenkerliftmapstartmapaugconsistencycheckconvert
MAGMA programs
4 ROBERT R. BRUNER
5. Generators
Name s n value precursor stemh0 1 0 (1)h1 1 1 (1) 0h2 1 3 (1) 1h3 1 7 (1) 3h4 1 15 (1) 7h5 1 31 (1) 15h6 1 63 (1) 31h7 1 127 (1) 63c0 3 8 (1)c1 3 19 (1) 8c2 3 41 (1) 19c3 3 85 (1) 41d0 4 14 (1) 5e0 4 17 (1)f0 4 18 (10) 7g 4 20 (1) 8d1 4 32 (1) 14p 4 33 (1)e1 4 38 (11) 17f1 4 40 (11) 18g2 4 44 (1) 20D3 4 61 (1)d2 4 68 (1) 32p′ 4 69 (1)p1 4 70 (11) 33e2 4 80 (11) 38f2 4 84 (10) 40g3 4 92 (1) 44D31 4 126 (10) 61
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 5
Name s n value precursor stemPh1 5 9 (1) 2Ph2 5 11 (1) 3n 5 31 (10) 13x 5 37 (1) 16D1 5 52 (1)H1 5 62 (100)n1 5 67 (10) 31Q3 5 67 (01) 31x1 5 79 (1) 37D11 5 109 (1) 52x5,77 5 125 (1) 60x5,80 5 128 (1)H11 5 129 (100) 62r 6 30 (1) 12q 6 32 (1) 13t 6 36 (1) 15y 6 38 (10) 16C 6 50 (1) 22G 6 54 (1) 24D2 6 58 (1) 26A′ 6 61 (11)A 6 61 (01)A′′ 6 64 (1) 29r1 6 66 (1) 30x6,47 6 71 (10)x6,53 6 76 (1) 35t1 6 78 (110) 36x6,68 6 85 (10)C1 6 106 (1) 50x6,89 6 108 (10) 51G1 6 114 (1) 54x6,94 6 124 (1) 59x6,97 6 126 (10) 60x6,99 6 127 (10)x6,101 6 128 (1000) 61x6,102 6 128 (0100) 61x6,107 6 132 (010) 63x6,110 6 134 (1) 64
6 ROBERT R. BRUNER
Name s n value precursor stemPc0 7 16 (1)i 7 23 (1) 8j 7 26 (1)k 7 29 (1) 11l 7 32 (1)m 7 35 (1) 14B1 7 46 (1)B2 7 48 (10)Q2 7 57 (1) 25B3 7 60 (1)x7,33 7 63 (100) 28x7,34 7 63 (010) 28x7,40 7 66 (10)x7,53 7 75 (1) 34m1 7 77 (1010) 35x7,57 7 77 (0100) 35x7,74 7 87 (1) 40x7,79 7 95 (10) 44x7,81 7 97 (10) 45x7,83 7 99 (100) 46x7,84 7 99 (010) 46x7,88 7 101 (010) 47x7,90 7 103 (10) 48x7,92 7 105 (1) 49x7,93 7 107 (10) 50x7,97 7 112 (1)x7,101 7 121 (10) 57x7,103 7 124 (100)x7,109 7 127 (10000) 60x7,110 7 127 (01000) 60x7,118 7 130 (1)x7,124 7 133 (100) 63
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 7
Name s n value precursor stemPd0 8 22 (1) 7Pe0 8 25 (1)N 8 46 (1) 19x8,32 8 62 (100) 27x8,33 8 62 (010) 27G21 8 68 (10) 30PD3 8 69 (01)x8,51 8 74 (10) 33x8,57 8 77 (100)x8,75 8 91 (1)x8,78 8 93 (10)x8,80 8 94 (1) 43x8,83 8 96 (10) 44x8,93 8 101 (10)x8,105 8 112 (100) 52x8,113 8 123 (1)x8,114 8 124 (1) 58x8,115 8 125 (10)x8,116 8 125 (01)x8,117 8 126 (100000) 59x8,118 8 126 (010000) 59x8,119 8 126 (001000) 59x8,120 8 126 (000100) 59x8,124 8 127 (0100000)x8,132 8 129 (01)x8,133 8 130 (10) 61x8,136 8 131 (0100)x8,139 8 132 (10000) 62x8,140 8 132 (01000) 62
8 ROBERT R. BRUNER
Name s n value precursor stemP 2h1 9 17 (1) 4P 2h2 9 19 (1) 5u 9 39 (1) 15v 9 42 (1)w 9 45 (1) 18B4 9 60 (10)X1 9 61 (1) 26x9,39 9 67 (10) 29x9,40 9 67 (01) 29x9,51 9 76 (10)x9,55 9 78 (10)x9,86 9 99 (10) 45x9,97 9 104 (10)x9,99 9 105 (1) 48x9,102 9 107 (1) 49x9,107 9 111 (1000) 51x9,109 9 111 (0010) 51x9,111 9 113 (1) 52x9,112 9 114 (10)x9,115 9 118 (10)x9,116 9 118 (01)x9,117 9 119 (1) 55x9,118 9 120 (1)x9,119 9 121 (10) 56x9,121 9 123 (10) 57x9,123 9 124 (100)x9,124 9 124 (010)x9,126 9 125 (10000) 58x9,129 9 125 (00010) 58x9,131 9 126 (100000)x9,145 9 129 (1000) 60x9,146 9 129 (0100) 60x9,154 9 131 (00100) 61
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 9
Name s n value precursor stemz 10 41 (1)x′ 10 53 (1)R1 10 54 (10) 22Q1 10 56 (01) 23B21 10 59 (1)x10,27 10 62 (100) 26x10,28 10 62 (010) 26x10,32 10 64 (10) 27B23 10 65 (1)B5 10 66 (10) 28PD2 10 66 (01) 28PA 10 69 (1)P 2h2
5 10 78 (11) 34Px6,53 10 84 (1) 37x10,60 10 87 (1)x10,63 10 90 (1) 40x10,65 10 92 (10) 41x10,67 10 93 (100)x10,70 10 94 (100) 42x10,73 10 95 (10)x10,76 10 97 (10)x10,82 10 100 (100) 45x10,100 10 109 (10)x10,102 10 110 (100) 50x10,107 10 112 (10) 51x10,109 10 113 (1)x10,113 10 117 (10)x10,114 10 117 (01)x10,116 10 118 (010) 54x10,118 10 119 (10)x10,120 10 120 (100) 55x10,124 10 123 (100)x10,127 10 124 (10000) 57x10,128 10 124 (01000) 57x10,132 10 125 (10000)x10,133 10 125 (01000)x10,137 10 126 (10000) 58x10,143 10 127 (010000)x10,148 10 128 (1000) 59x10,149 10 128 (0100) 59x10,152 10 129 (100)x10,155 10 130 (10000) 60
10 ROBERT R. BRUNER
Name s n value precursor stemP 2c0 11 24 (1)Pj 11 34 (1)x11,35 11 67 (10) 28x11,59 11 89 (1) 39x11,61 11 91 (10) 40x11,80 11 101 (010) 45x11,91 11 108 (10)x11,101 11 115 (1) 52x11,103 11 117 (100) 53x11,106 11 118 (100)x11,109 11 119 (1000) 54x11,113 11 120 (100)x11,116 11 121 (1) 55x11,118 11 122 (01)x11,119 11 123 (10000) 56x11,120 11 123 (01000) 56x11,124 11 124 (10000)x11,125 11 124 (01000)x11,126 11 124 (00100)x11,134 11 126 (100000)x11,147 11 129 (100000) 59x11,148 11 129 (010000) 59
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 11
Name s n value precursor stemP 2d0 12 30 (1) 9P 2e0 12 33 (1)x12,37 12 71 (10)P 2D3 12 77 (10)x12,44 12 80 (1) 34x12,45 12 81 (1)x12,48 12 86 (10) 37x12,51 12 88 (100) 38x12,55 12 90 (1) 39x12,58 12 92 (10) 40x12,60 12 93 (10)x12,64 12 96 (10) 42x12,78 12 103 (10)x12,80 12 104 (100) 46x12,85 12 107 (1)x12,86 12 109 (10)x12,93 12 114 (01) 51x12,96 12 117 (1000)x12,100 12 118 (100) 53x12,106 12 120 (10000) 54x12,107 12 120 (01000) 54x12,116 12 123 (100)x12,124 12 125 (10000)x12,125 12 125 (01000)x12,137 12 128 (10000) 58x12,140 12 128 (00010) 58
12 ROBERT R. BRUNER
Name s n value precursor stemP 3h1 13 25 (1) 6P 3h2 13 27 (1) 7Q 13 47 (10) 17Pu 13 47 (01) 17Pv 13 50 (1)R2 13 65 (01) 26P 2D1 13 68 (10)W1 13 69 (1) 28x13,34 13 71 (010) 29x13,35 13 71 (001) 29x13,42 13 79 (1) 33x13,46 13 85 (1) 36x13,73 13 100 (100)x13,85 13 107 (1) 47x13,87 13 109 (1) 48x13,88 13 110 (10)x13,91 13 112 (10)x13,93 13 113 (10) 50Px9,99 13 113 (01) 50x13,95 13 115 (10) 51x13,97 13 116 (100)x13,113 13 122 (100)x13,116 13 123 (100) 55x13,117 13 123 (010) 55x13,132 13 127 (01000) 57
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 13
Name s n value precursor stemPQ1 14 64 (10) 25x14,42 14 80 (10) 33x14,46 14 84 (1) 35P 3h2
5 14 86 (10) 36x14,67 14 98 (100) 42x14,74 14 102 (10) 44x14,79 14 106 (10) 46x14,82 14 108 (1) 47y14,83 14 109 (11)x14,84 14 109 (01)x14,91 14 112 (01) 49x14,104 14 118 (1000) 52Px10,102 14 118 (1111) 52x14,108 14 119 (10)x14,110 14 120 (10) 53Px10,109 14 121 (1)x14,117 14 124 (10000) 55x14,118 14 124 (01000) 55Px10,113 14 125 (111)x14,126 14 126 (0100) 56
14 ROBERT R. BRUNER
Name s n value precursor stemP 3c0 15 32 (1)P 2i 15 39 (1) 12P 2j 15 42 (1)x15,41 15 83 (1) 34x15,42 15 84 (10)x15,43 15 84 (01)x15,47 15 87 (10) 36x15,56 15 94 (1)x15,58 15 96 (10)x15,65 15 102 (1)x15,68 15 105 (10) 45x15,74 15 108 (01)x15,78 15 110 (10)x15,81 15 111 (0100) 48x15,82 15 111 (0010) 48x15,90 15 114 (10)x15,96 15 116 (0010)x15,97 15 116 (0001)x15,98 15 117 (100) 51x15,103 15 119 (100) 52x15,108 15 122 (10)x15,109 15 122 (01)x15,110 15 123 (100) 54x15,113 15 124 (1000)x15,114 15 124 (0100)x15,117 15 125 (10000) 55x15,119 15 125 (00100) 55
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 15
Name s n value precursor stemP 3d0 16 38 (1) 11P 3e0 16 41 (1)x16,32 16 76 (1) 30x16,33 16 77 (10)x16,35 16 79 (1)x16,37 16 80 (01) 32x16,38 16 82 (1) 33x16,42 16 85 (100)x16,48 16 88 (1) 36x16,54 16 94 (1) 39Px12,60 16 101 (1)x16,77 16 110 (1000) 47x16,78 16 110 (0100) 47Px12,80 16 112 (0100) 48Px12,85 16 115 (10)x16,95 16 116 (1000) 50Px12,86 16 117 (00100)x16,109 16 121 (010)x16,112 16 122 (010) 53x16,117 16 124 (10000) 54P 4h1 17 33 (1) 8P 4h2 17 35 (1) 9P 2u 17 55 (1) 19P 2v 17 58 (1)R1 17 70 (10)PR2 17 73 (10) 28x17,50 17 87 (1) 35x17,52 17 90 (10)x17,57 17 93 (1) 38x17,76 17 108 (100)x17,79 17 109 (10) 46x17,80 17 109 (01) 46x17,93 17 115 (10) 49x17,94 17 115 (01) 49Px13,87 17 117 (1000) 50P 2x9,97 17 120 (100)P 2x9,99 17 121 (0101) 52
16 ROBERT R. BRUNER
Name s n value precursor stemx18,20 18 69 (10)P 2Q1 18 72 (10) 27x18,50 18 89 (1)x18,55 18 92 (1) 37x18,57 18 95 (1)x18,60 18 98 (1) 40x18,63 18 101 (10)x18,68 18 104 (010) 43x18,72 18 107 (10)x18,77 18 110 (1000) 46x18,78 18 110 (0100) 46x18,83 18 112 (010) 47x18,85 18 113 (10)P 2x10,76 18 113 (01)x18,87 18 114 (100) 48Px14,79 18 114 (011) 48Px14,82 18 116 (1000) 49Px14,84 18 117 (01)P 4c0 19 40 (1)P 3j 19 50 (1)x19,49 19 94 (1)x19,58 19 102 (10)P 2x11,61 19 107 (101) 44Px15,65 19 110 (0100)x19,86 19 115 (10) 48Px15,78 19 118 (1)P 4d0 20 46 (1) 13P 4e0 20 49 (1)Px16,35 20 87 (1)x20,91 20 119 (1000)P 2x12,80 20 120 (100) 50
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 17
Name s n value precursor stemP 5h1 21 41 (1) 10P 5h2 21 43 (1) 11P 3u 21 63 (10) 21P 3v 21 66 (1)P 2R2 21 81 (10) 30x21,43 21 95 (10) 37Px17,50 21 95 (01) 37Px17,52 21 98 (1)x21,84 21 117 (01) 48x21,87 21 119 (010) 49P 3Q1 22 80 (10) 29x22,39 22 94 (1) 36Px18,50 22 97 (1)Px18,55 22 100 (1) 39Px18,68 22 112 (10) 45x22,71 22 113 (100)x22,78 22 116 (010) 47P 5c0 23 48 (1)P 4i 23 55 (1) 16P 4j 23 58 (1)P 5d0 24 54 (1) 15P 5e0 24 57 (1)P 2x16,32 24 92 (1) 34P 2x16,35 24 95 (11)P 6h1 25 49 (1) 12P 6h2 25 51 (1) 13P 4u 25 71 (1) 23P 4v 25 74 (1)x25,24 25 86 (10)P 3R2 25 89 (11) 32P 2x17,50 25 103 (1) 39P 2x17,52 25 106 (10)P 4x′ 26 85 (10)P 4Q1 26 88 (11) 31P 2x18,50 26 105 (1)
18 ROBERT R. BRUNER
Name s n value precursor stemP 6c0 27 56 (1)P 5j 27 66 (1)P 6d0 28 62 (1) 17P 6e0 28 65 (1)P 3x16,35 28 103 (1)P 7h1 29 57 (1) 14P 7h2 29 59 (1) 15P 4Q 29 79 (11) 25P 5u 29 79 (01) 25P 5v 29 82 (1)P 4R2 29 97 (10) 34P 3x17,50 29 111 (1) 41P 5Q1 30 96 (11) 33P 7c0 31 64 (1)P 6i 31 71 (1) 20P 6j 31 74 (1)P 7d0 32 70 (1) 19P 7e0 32 73 (1)P 4x16,32 32 108 (1) 38P 8h1 33 65 (1) 16P 8h2 33 67 (1) 17P 6u 33 87 (1) 27P 6v 33 90 (1)P 4R1 33 102 (11)P 5R2 33 105 (10) 36P 4x18,20 34 101 (11)P 6Q1 34 104 (11) 35P 8c0 35 72 (1)P 7j 35 82 (1)P 8d0 36 78 (1) 21P 8e0 36 81 (1)P 9h1 37 73 (1) 18P 9h2 37 75 (1) 19P 7u 37 95 (10) 29P 7v 37 98 (1)P 9c0 39 80 (1)P 8i 39 87 (1) 24P 8j 39 90 (1)
6. The tables
Stem 11 (1) h1
Stem 22 (1) h2
1
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 19
Stem 31 (1) h2
2 (1) h0h2
3 (1) h20h2 = h3
1
Stem 4
Stem 5
Stem 62 (1) h2
2
Stem 71 (1) h3
2 (1) h0h3
3 (1) h20h3
4 (1) h30h3
Stem 82 (1) h1h3
3 (1) c0
Stem 93 (1) h2
1h3 = h32
4 (1) h1c0
5 (1) Ph1
Stem 106 (1) h1Ph1
Stem 115 (1) Ph2
6 (1) h0Ph2
7 (1) h20Ph2 = h2
1Ph1
Stem 12
Stem 13
Stem 142 (1) h2
3
3 (1) h0h23
4 (1) d0
5 (1) h0d0
6 (1) h20d0 = h2Ph2
20 ROBERT R. BRUNER
Stem 151 (1) h4
2 (1) h0h4
3 (1) h20h4
4 (1) h30h4
5 (10) h1d0
(01) h40h4
6 (1) h50h4
7 (1) h60h4
8 (1) h70h4
Stem 162 (1) h1h4
6 (1) h21d0 = h3Ph1 = c2
0
7 (1) Pc0
Stem 173 (1) h2
1h4
4 (1) e0
5 (1) h0e0 = h2d0
6 (1) h20e0 = h0h2d0
7 (1) h30e0 = h2
0h2d0 = h31d0 = h1h3Ph1 = h1c
20 = h2
2Ph2
8 (1) h1Pc0 = c0Ph1
9 (1) P 2h1
Stem 182 (1) h2h4
3 (1) h0h2h4
4 (10) f0
(01) h20h2h4 = h3
1h4
5 (1) h0f0 = h1e0
10 (1) h1P2h1 = Ph2
1
Stem 193 (1) c1
9 (1) P 2h2
10 (1) h0P2h2
11 (1) h20P
2h2 = h21P
2h1 = h1Ph21
Stem 204 (1) g
5 (1) h0g = h2e0
6 (1) h20g = h0h2e0 = h2
2d0
Stem 213 (1) h2
2h4 = h33
5 (1) h1g = h2f0
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 21
Stem 224 (1) h2c1
8 (1) Pd0
9 (1) h0Pd0
10 (1) h20Pd0 = h2P
2h2 = Ph22
Stem 234 (1) h4c0
5 (1) h2g
6 (1) h0h2g = h22e0
7 (1) i
8 (1) h0i
9 (01) h20i
(11) h1Pd0 = d0Ph1
10 (1) h30i
11 (1) h40i
12 (1) h50i
Stem 245 (1) h1h4c0 = h3e0
10 (1) h21Pd0 = h1d0Ph1 = h3P
2h1 = c0Pc0
11 (1) P 2c0
Stem 258 (1) Pe0
9 (1) h0Pe0 = h2Pd0 = d0Ph2
10 (1) h20Pe0 = h0h2Pd0 = h0d0Ph2
11 (1) h30Pe0 = h2
0h2Pd0 = h20d0Ph2 = h3
1Pd0 = h21d0Ph1 =
h1h3P2h1 = h1c0Pc0 = h2
2P2h2 = h2Ph2
2 = h3Ph21 = c2
0Ph1
12 (1) h1P2c0 = c0P
2h1 = Ph1Pc0
13 (1) P 3h1
Stem 266 (1) h2
2g = h4Ph2
7 (1) j
8 (1) h0j = h2i
9 (1) h20j = h0h2i = h1Pe0 = e0Ph1
14 (1) h1P3h1 = Ph1P
2h1
Stem 2713 (1) P 3h2
14 (1) h0P3h2
15 (1) h20P
3h2 = h21P
3h1 = h1Ph1P2h1 = Ph3
1
Stem 288 (1) d2
0
9 (1) h0d20 = h2Pe0 = e0Ph2
10 (1) h20d
20 = h0h2Pe0 = h0e0Ph2 = h2
2Pd0 = h2d0Ph2
22 ROBERT R. BRUNER
Stem 297 (1) k
8 (1) h0k = h2j
9 (1) h20k = h0h2j = h1d
20 = h2
2i = f0Ph2 = gPh1
Stem 302 (1) h2
4
3 (1) h0h24
4 (1) h20h
24
5 (1) h30h
24
6 (1) r
7 (1) h0r
8 (1) h20r = h3i
9 (1) h30r = h0h3i
10 (1) h40r = h2
0h3i
11 (1) h50r = h3
0h3i = c0Pd0 = d0Pc0
12 (1) P 2d0
13 (1) h0P2d0
14 (1) h20P
2d0 = h2P3h2 = Ph2P
2h2
Stem 311 (1) h5
2 (1) h0h5
3 (10) h1h24
(01) h20h5
4 (1) h30h5
5 (10) n
(01) h40h5
6 (1) h50h5
7 (1) h60h5
8 (10) d0e0
(01) h70h5
9 (10) h0d0e0 = h2d20 = gPh2
(01) h80h5
10 (10) h20d0e0 = h0h2d
20 = h0gPh2 = h2
2Pe0 = h2e0Ph2 = c0i
(01) h90h5
11 (1) h100 h5
12 (1) h110 h5
13 (10) h1P2d0 = d0P
2h1 = Ph1Pd0
(01) h120 h5
14 (1) h130 h5
15 (1) h140 h5
16 (1) h150 h5
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 23
Stem 322 (1) h1h5
4 (1) d1
6 (1) q
7 (1) l
8 (1) h0l = h2k = d0f0
9 (1) h20l = h0h2k = h0d0f0 = h1d0e0 = h2
2j
14 (1) h21P
2d0 = h1d0P2h1 = h1Ph1Pd0 = h3P
3h1 = c0P2c0 = d0Ph2
1 =Pc2
0
15 (1) P 3c0
Stem 333 (1) h2
1h5
4 (1) p
5 (1) h0p = h1d1
7 (1) h1q = h2r
12 (1) P 2e0
13 (1) h0P2e0 = h2P
2d0 = d0P2h2 = Ph2Pd0
14 (1) h20P
2e0 = h0h2P2d0 = h0d0P
2h2 = h0Ph2Pd0
15 (1) h30P
2e0 = h20h2P
2d0 = h20d0P
2h2 = h20Ph2Pd0 = h3
1P2d0 =
h21d0P
2h1 = h21Ph1Pd0 = h1h3P
3h1 = h1c0P2c0 = h1d0Ph2
1 =h1Pc2
0 = h22P
3h2 = h2Ph2P2h2 = h3Ph1P
2h1 = c20P
2h1 =c0Ph1Pc0 = Ph3
2
16 (1) h1P3c0 = c0P
3h1 = Ph1P2c0 = Pc0P
2h1
17 (1) P 4h1
Stem 342 (1) h2h5
3 (1) h0h2h5
4 (1) h20h2h5 = h3
1h5
6 (1) h2n
8 (1) d0g = e20
9 (1) h0d0g = h0e20 = h2d0e0
10 (1) h20d0g = h2
0e20 = h0h2d0e0 = h2
2d20 = h2gPh2 = h4P
2h2 = c0j
11 (1) Pj
12 (1) h0Pj = Ph2i
13 (1) h20Pj = h0Ph2i = h1P
2e0 = e0P2h1 = Ph1Pe0
18 (1) h1P4h1 = Ph1P
3h1 = P 2h21
Stem 355 (1) h2d1 = h4g
7 (1) m
8 (1) h0m = h2l = e0f0
9 (1) h20m = h0h2l = h0e0f0 = h1d0g = h1e
20 = h2
2k = h2d0f0
17 (1) P 4h2
18 (1) h0P4h2
19 (1) h20P
4h2 = h21P
4h1 = h1Ph1P3h1 = h1P
2h21 = Ph2
1P2h1
24 ROBERT R. BRUNER
Stem 366 (1) t
12 (1) d0Pd0
13 (1) h0d0Pd0 = h2P2e0 = e0P
2h2 = Ph2Pe0
14 (1) h20d0Pd0 = h0h2P
2e0 = h0e0P2h2 = h0Ph2Pe0 = h2
2P2d0 =
h2d0P2h2 = h2Ph2Pd0 = d0Ph2
2
Stem 373 (1) h2
2h5
5 (1) x
6 (1) h0x
7 (10) h1t = h22n = c1f0
(01) h20x
(11) h3r
8 (10) e0g
(01) h30x = h0h3r
9 (1) h40x = h2
0h3r = h0e0g = h2d0g = h2e20 = h2
3i = h4Pd0
10 (1) h50x = h3
0h3r = h20e0g = h0h2d0g = h0h2e
20 = h0h
23i = h0h4Pd0 =
h22d0e0 = c0k
11 (1) d0i
12 (1) h0d0i = h2Pj = Ph2j
13 (1) h20d0i = h0h2Pj = h0Ph2j = h1d0Pd0 = h2Ph2i = d2
0Ph1 =f0P
2h2 = gP 2h1
Stem 382 (1) h3h5
3 (1) h0h3h5
4 (01) h20h3h5
(11) e1
5 (1) h30h3h5
6 (10) y
(01) h1x = h22d1 = h2h4g = h3n = c2
1
7 (1) h0y
8 (1) h20y = h2m = f0g
9 (1) h30y = h0h2m = h0f0g = h1e0g = h2
2l = h2e0f0 = c0r
16 (1) P 3d0
17 (1) h0P3d0
18 (1) h20P
3d0 = h2P4h2 = Ph2P
3h2 = P 2h22
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 25
Stem 393 (1) h1h3h5
4 (1) h5c0
5 (1) h1e1 = h3d1
7 (1) h1y = h2t = h3q = c1g
9 (1) u
12 (1) d0Pe0 = e0Pd0
13 (1) h0d0Pe0 = h0e0Pd0 = h2d0Pd0 = d20Ph2 = gP 2h2
14 (1) h20d0Pe0 = h2
0e0Pd0 = h0h2d0Pd0 = h0d20Ph2 = h0gP 2h2 =
h22P
2e0 = h2e0P2h2 = h2Ph2Pe0 = e0Ph2
2 = Pc0i
15 (1) P 2i
16 (1) h0P2i
17 (01) h20P
2i
(11) h1P3d0 = d0P
3h1 = Ph1P2d0 = Pd0P
2h1
18 (1) h30P
2i
19 (1) h40P
2i
20 (1) h50P
2i
Stem 404 (01) h2
1h3h5 = h32h5
(11) f1
5 (01) h0f1 = h3p
(11) h1h5c0
6 (10) h5Ph1
(01) h20f1 = h0h3p = h2
1e1 = h1h3d1
8 (1) g2
10 (1) h1u
11 (1) d0j = e0i
12 (1) h0d0j = h0e0i = h2d0i = f0Pd0 = Ph2k
13 (1) h20d0j = h2
0e0i = h0h2d0i = h0f0Pd0 = h0Ph2k = h1d0Pe0 =h1e0Pd0 = h2
2Pj = h2Ph2j = d0e0Ph1
18 (1) h21P
3d0 = h1d0P3h1 = h1Ph1P
2d0 = h1Pd0P2h1 = h3P
4h1 =c0P
3c0 = d0Ph1P2h1 = Ph2
1Pd0 = Pc0P2c0
19 (1) P 4c0
26 ROBERT R. BRUNER
Stem 413 (1) c2
4 (1) h0c2
5 (1) h20c2 = h1f1 = h2e1
7 (1) h1h5Ph1
10 (1) z
11 (1) h0z = h21u = f0i = Ph1q = Ph2r
16 (1) P 3e0
17 (1) h0P3e0 = h2P
3d0 = d0P3h2 = Ph2P
2d0 = Pd0P2h2
18 (1) h20P
3e0 = h0h2P3d0 = h0d0P
3h2 = h0Ph2P2d0 = h0Pd0P
2h2
19 (1) h30P
3e0 = h20h2P
3d0 = h20d0P
3h2 = h20Ph2P
2d0 = h20Pd0P
2h2 =h3
1P3d0 = h2
1d0P3h1 = h2
1Ph1P2d0 = h2
1Pd0P2h1 = h1h3P
4h1 =h1c0P
3c0 = h1d0Ph1P2h1 = h1Ph2
1Pd0 = h1Pc0P2c0 =
h22P
4h2 = h2Ph2P3h2 = h2P
2h22 = h3Ph1P
3h1 = h3P2h2
1 =c20P
3h1 = c0Ph1P2c0 = c0Pc0P
2h1 = d0Ph31 = Ph1Pc2
0 =Ph2
2P2h2
20 (1) h1P4c0 = c0P
4h1 = Ph1P3c0 = Pc0P
3h1 = P 2h1P2c0
21 (1) P 5h1
Stem 426 (1) h5Ph2
7 (1) h0h5Ph2
8 (1) h20h5Ph2 = h2
1h5Ph1
9 (1) v
12 (1) d30 = e0Pe0 = gPd0
13 (1) h0d30 = h0e0Pe0 = h0gPd0 = h2d0Pe0 = h2e0Pd0 = d0e0Ph2
14 (1) h20d
30 = h2
0e0Pe0 = h20gPd0 = h0h2d0Pe0 = h0h2e0Pd0 =
h0d0e0Ph2 = h22d0Pd0 = h2d
20Ph2 = h2gP 2h2 = h4P
3h2 =c0Pj = gPh2
2 = Pc0j
15 (1) P 2j
16 (1) h0P2j = h2P
2i = iP 2h2
17 (1) h20P
2j = h0h2P2i = h0iP
2h2 = h1P3e0 = e0P
3h1 = Ph1P2e0 =
Pe0P2h1
22 (1) h1P5h1 = Ph1P
4h1 = P 2h1P3h1
Stem 4311 (1) d0k = e0j = gi
12 (1) h0d0k = h0e0j = h0gi = h2d0j = h2e0i = f0Pe0 = Ph2l
13 (1) h20d0k = h2
0e0j = h20gi = h0h2d0j = h0h2e0i = h0f0Pe0 =
h0Ph2l = h1d30 = h1e0Pe0 = h1gPd0 = h2
2d0i = h2f0Pd0 =h2Ph2k = d0f0Ph2 = d0gPh1 = e2
0Ph1
21 (1) P 5h2
22 (1) h0P5h2
23 (1) h20P
5h2 = h21P
5h1 = h1Ph1P4h1 = h1P
2h1P3h1 = Ph2
1P3h1 =
Ph1P2h2
1
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 27
Stem 444 (1) g2
5 (1) h0g2
6 (1) h20g2 = h3x
10 (1) d0r
16 (1) d0P2d0 = Pd2
0
17 (1) h0d0P2d0 = h0Pd2
0 = h2P3e0 = e0P
3h2 = Ph2P2e0 = Pe0P
2h2
18 (1) h20d0P
2d0 = h20Pd2
0 = h0h2P3e0 = h0e0P
3h2 = h0Ph2P2e0 =
h0Pe0P2h2 = h2
2P3d0 = h2d0P
3h2 = h2Ph2P2d0 = h2Pd0P
2h2 =d0Ph2P
2h2 = Ph22Pd0
Stem 453 (1) h2
3h5 = h34
4 (1) h0h23h5 = h0h
34
5 (10) h5d0
(01) h1g2 = h3e1
6 (1) h0h5d0
7 (1) h20h5d0 = h2h5Ph2 = h3y
9 (1) w
12 (1) d20e0 = gPe0
15 (1) iPd0
16 (1) h0iPd0 = h2P2j = Ph2Pj = jP 2h2
17 (1) h20iPd0 = h0h2P
2j = h0Ph2Pj = h0jP2h2 = h1d0P
2d0 =h1Pd2
0 = h22P
2i = h2iP2h2 = d2
0P2h1 = d0Ph1Pd0 = f0P
3h2 =gP 3h1 = Ph2
2i
Stem 466 (1) h1h5d0
7 (1) B1
8 (1) N
11 (1) d0l = e0k = gj
14 (1) i2
15 (1) h0i2
16 (1) h20i
2 = h3P2i
17 (1) h30i
2 = h0h3P2i
18 (1) h40i
2 = h20h3P
2i
19 (1) h50i
2 = h30h3P
2i = c0P3d0 = d0P
3c0 = Pc0P2d0 = Pd0P
2c0
20 (1) P 4d0
21 (1) h0P4d0
22 (1) h20P
4d0 = h2P5h2 = Ph2P
4h2 = P 2h2P3h2
28 ROBERT R. BRUNER
Stem 475 (1) h2g2 = h3f1
6 (1) h0h2g2 = h0h3f1 = h23p
7 (1) h21h5d0 = h3h5Ph1 = h4q = h5c
20
8 (10) h5Pc0
(01) h1B1
10 (1) e0r
13 (10) Q
(01) Pu
14 (1) h0Q
15 (1) h20Q
16 (10) d0P2e0 = e0P
2d0 = Pd0Pe0
(01) h30Q
17 (01) h40Q
(11) h0d0P2e0 = h0e0P
2d0 = h0Pd0Pe0 = h2d0P2d0 = h2Pd2
0 =d20P
2h2 = d0Ph2Pd0 = gP 3h2
18 (01) h50Q
(11) h20d0P
2e0 = h20e0P
2d0 = h20Pd0Pe0 = h0h2d0P
2d0 = h0h2Pd20 =
h0d20P
2h2 = h0d0Ph2Pd0 = h0gP 3h2 = h22P
3e0 = h2e0P3h2 =
h2Ph2P2e0 = h2Pe0P
2h2 = c0P2i = e0Ph2P
2h2 = Ph22Pe0 =
iP 2c0
19 (1) h60Q
20 (1) h70Q
21 (10) h1P4d0 = d0P
4h1 = Ph1P3d0 = Pd0P
3h1 = P 2h1P2d0
(01) h80Q
22 (1) h90Q
23 (1) h100 Q
24 (1) h110 Q
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 29
Stem 484 (1) h3c2
5 (1) h5e0
6 (1) h0h5e0 = h2h5d0
7 (10) B2
(01) h20h5e0 = h0h2h5d0
8 (1) h0B2
9 (10) h1h5Pc0 = h5c0Ph1
(01) h20B2 = h2
1B1
12 (1) d20g = d0e
20
14 (1) h1Q = h1Pu = Ph1u
15 (1) d0Pj = iPe0 = jPd0
16 (1) h0d0Pj = h0iPe0 = h0jPd0 = h2iPd0 = d0Ph2i = f0P2d0 =
kP 2h2
17 (1) h20d0Pj = h2
0iPe0 = h20jPd0 = h0h2iPd0 = h0d0Ph2i =
h0f0P2d0 = h0kP 2h2 = h1d0P
2e0 = h1e0P2d0 = h1Pd0Pe0 =
h22P
2j = h2Ph2Pj = h2jP2h2 = d0e0P
2h1 = d0Ph1Pe0 =e0Ph1Pd0 = Ph2
2j
22 (1) h21P
4d0 = h1d0P4h1 = h1Ph1P
3d0 = h1Pd0P3h1 =
h1P2h1P
2d0 = h3P5h1 = c0P
4c0 = d0Ph1P3h1 = d0P
2h21 =
Ph21P
2d0 = Ph1Pd0P2h1 = Pc0P
3c0 = P 2c20
23 (1) P 5c0
Stem 495 (1) h5f0
6 (1) h0h5f0 = h1h5e0
11 (1) d0m = e0l = gk
14 (1) ij
15 (1) h0ij = h21Q = h2
1Pu = h1Ph1u = h2i2 = rP 2h2 = qP 2h1
20 (1) P 4e0
21 (1) h0P4e0 = h2P
4d0 = d0P4h2 = Ph2P
3d0 = Pd0P3h2 = P 2h2P
2d0
22 (1) h20P
4e0 = h0h2P4d0 = h0d0P
4h2 = h0Ph2P3d0 = h0Pd0P
3h2 =h0P
2h2P2d0
23 (1) h30P
4e0 = h20h2P
4d0 = h20d0P
4h2 = h20Ph2P
3d0 = h20Pd0P
3h2 =h2
0P2h2P
2d0 = h31P
4d0 = h21d0P
4h1 = h21Ph1P
3d0 =h2
1Pd0P3h1 = h2
1P2h1P
2d0 = h1h3P5h1 = h1c0P
4c0 =h1d0Ph1P
3h1 = h1d0P2h2
1 = h1Ph21P
2d0 = h1Ph1Pd0P2h1 =
h1Pc0P3c0 = h1P
2c20 = h2
2P5h2 = h2Ph2P
4h2 = h2P2h2P
3h2 =h3Ph1P
4h1 = h3P2h1P
3h1 = c20P
4h1 = c0Ph1P3c0 =
c0Pc0P3h1 = c0P
2h1P2c0 = d0Ph2
1P2h1 = Ph3
1Pd0 =Ph1Pc0P
2c0 = Ph22P
3h2 = Ph2P2h2
2 = Pc20P
2h1
24 (1) h1P5c0 = c0P
5h1 = Ph1P4c0 = Pc0P
4h1 = P 2h1P3c0 =
P 2c0P3h1
25 (1) P 6h1
30 ROBERT R. BRUNER
Stem 504 (1) h5c1
6 (1) C
10 (1) gr
13 (1) Pv
16 (1) d20Pd0 = e0P
2e0 = gP 2d0 = Pe20
17 (1) h0d20Pd0 = h0e0P
2e0 = h0gP 2d0 = h0Pe20 = h2d0P
2e0 =h2e0P
2d0 = h2Pd0Pe0 = d0e0P2h2 = d0Ph2Pe0 = e0Ph2Pd0
18 (1) h20d
20Pd0 = h2
0e0P2e0 = h2
0gP 2d0 = h20Pe2
0 = h0h2d0P2e0 =
h0h2e0P2d0 = h0h2Pd0Pe0 = h0d0e0P
2h2 = h0d0Ph2Pe0 =h0e0Ph2Pd0 = h2
2d0P2d0 = h2
2Pd20 = h2d
20P
2h2 = h2d0Ph2Pd0 =h2gP 3h2 = h4P
4h2 = c0P2j = d2
0Ph22 = gPh2P
2h2 = Pc0Pj =jP 2c0
19 (1) P 3j
20 (1) h0P3j = Ph2P
2i = iP 3h2
21 (1) h20P
3j = h0Ph2P2i = h0iP
3h2 = h1P4e0 = e0P
4h1 = Ph1P3e0 =
Pe0P3h1 = P 2h1P
2e0
26 (1) h1P6h1 = Ph1P
5h1 = P 2h1P4h1 = P 3h2
1
Stem 515 (1) h3g2 = h5g
6 (1) h0h3g2 = h0h5g = h2h5e0
7 (1) h20h3g2 = h2
0h5g = h0h2h5e0 = h22h5d0 = h2
3x
8 (1) h2B2
9 (1) gn
12 (1) d0e0g = e30
15 (1) d20i = e0Pj = jPe0 = kPd0
16 (1) h0d20i = h0e0Pj = h0jPe0 = h0kPd0 = h2d0Pj = h2iPe0 =
h2jPd0 = d0Ph2j = e0Ph2i = f0P2e0 = lP 2h2
17 (1) h20d
20i = h2
0e0Pj = h20jPe0 = h2
0kPd0 = h0h2d0Pj = h0h2iPe0 =h0h2jPd0 = h0d0Ph2j = h0e0Ph2i = h0f0P
2e0 = h0lP2h2 =
h1d20Pd0 = h1e0P
2e0 = h1gP 2d0 = h1Pe20 = h2
2iPd0 =h2d0Ph2i = h2f0P
2d0 = h2kP 2h2 = d30Ph1 = d0f0P
2h2 =d0gP 2h1 = e2
0P2h1 = e0Ph1Pe0 = f0Ph2Pd0 = gPh1Pd0 = Ph2
2k
25 (1) P 6h2
26 (1) h0P6h2
27 (1) h20P
6h2 = h21P
6h1 = h1Ph1P5h1 = h1P
2h1P4h1 = h1P
3h21 =
Ph21P
4h1 = Ph1P2h1P
3h1 = P 2h31
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 31
Stem 525 (1) D1
6 (1) h1h3g2 = h1h5g = h2h5f0 = h23e1
8 (1) gd1
11 (1) e0m = gl
14 (1) rPd0 = ik = j2
20 (1) d0P3d0 = Pd0P
2d0
21 (1) h0d0P3d0 = h0Pd0P
2d0 = h2P4e0 = e0P
4h2 = Ph2P3e0 =
Pe0P3h2 = P 2h2P
2e0
22 (1) h20d0P
3d0 = h20Pd0P
2d0 = h0h2P4e0 = h0e0P
4h2 =h0Ph2P
3e0 = h0Pe0P3h2 = h0P
2h2P2e0 = h2
2P4d0 =
h2d0P4h2 = h2Ph2P
3d0 = h2Pd0P3h2 = h2P
2h2P2d0 =
d0Ph2P3h2 = d0P
2h22 = Ph2
2P2d0 = Ph2Pd0P
2h2
Stem 535 (1) h2h5c1 = h4e1
7 (1) h2C
9 (1) h5Pd0
10 (1) x′
11 (1) h0x′
12 (1) h20x
′
13 (10) d0u
(01) h30x
′
(11) ri
14 (1) h40x
′ = h0ri
15 (1) h50x
′ = h20ri = h3i
2
16 (01) h60x
′ = h30ri = h0h3i
2
(11) d20Pe0 = d0e0Pd0 = gP 2e0
17 (1) h70x
′ = h40ri = h2
0h3i2 = h0d
20Pe0 = h0d0e0Pd0 = h0gP 2e0 =
h2d20Pd0 = h2e0P
2e0 = h2gP 2d0 = h2Pe20 = h2
3P2i = h4P
3d0 =d30Ph2 = d0gP 2h2 = e2
0P2h2 = e0Ph2Pe0 = gPh2Pd0
18 (1) h80x
′ = h50ri = h3
0h3i2 = h2
0d20Pe0 = h2
0d0e0Pd0 = h20gP 2e0 =
h0h2d20Pd0 = h0h2e0P
2e0 = h0h2gP 2d0 = h0h2Pe20 =
h0h23P
2i = h0h4P3d0 = h0d
30Ph2 = h0d0gP 2h2 = h0e
20P
2h2 =h0e0Ph2Pe0 = h0gPh2Pd0 = h2
2d0P2e0 = h2
2e0P2d0 =
h22Pd0Pe0 = h2d0e0P
2h2 = h2d0Ph2Pe0 = h2e0Ph2Pd0 =c0iPd0 = d0e0Ph2
2 = d0Pc0i = kP 2c0
19 (1) d0P2i = iP 2d0
20 (1) h0d0P2i = h0iP
2d0 = h2P3j = Ph2P
2j = jP 3h2 = P 2h2Pj
21 (1) h20d0P
2i = h20iP
2d0 = h0h2P3j = h0Ph2P
2j = h0jP3h2 =
h0P2h2Pj = h1d0P
3d0 = h1Pd0P2d0 = h2Ph2P
2i = h2iP3h2 =
d20P
3h1 = d0Ph1P2d0 = d0Pd0P
2h1 = f0P4h2 = gP 4h1 =
Ph1Pd20 = Ph2iP
2h2
32 ROBERT R. BRUNER
Stem 546 (1) G
8 (1) h5i
9 (1) h0h5i
10 (10) R1
(01) h20h5i = h1h5Pd0 = h4u = h5d0Ph1 = c0B1
11 (10) h1x′
(01) h0R1
12 (10) d0g2 = e2
0g
(01) h20R1
13 (1) h30R1
14 (1) h40R1 = h3Q
15 (01) h50R1 = h0h3Q
(11) d20j = d0e0i = gPj = kPe0 = lPd0
16 (1) h60R1 = h2
0h3Q = h0d20j = h0d0e0i = h0gPj = h0kPe0 =
h0lPd0 = h2d20i = h2e0Pj = h2jPe0 = h2kPd0 = d0f0Pd0 =
d0Ph2k = e0Ph2j = gPh2i = mP 2h2
17 (1) h70R1 = h3
0h3Q = h20d
20j = h2
0d0e0i = h20gPj = h2
0kPe0 =h2
0lPd0 = h0h2d20i = h0h2e0Pj = h0h2jPe0 = h0h2kPd0 =
h0d0f0Pd0 = h0d0Ph2k = h0e0Ph2j = h0gPh2i = h0mP 2h2 =h1d
20Pe0 = h1d0e0Pd0 = h1gP 2e0 = h2
2d0Pj = h22iPe0 =
h22jPd0 = h2d0Ph2j = h2e0Ph2i = h2f0P
2e0 = h2lP2h2 = c0i
2 =d20e0Ph1 = e0f0P
2h2 = e0gP 2h1 = f0Ph2Pe0 = gPh1Pe0 =Ph2
2l = rP 2c0
24 (1) P 5d0
25 (1) h0P5d0
26 (1) h20P
5d0 = h2P6h2 = Ph2P
5h2 = P 2h2P4h2 = P 3h2
2
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 33
Stem 557 (1) h1G
11 (1) gm
12 (1) h21x
′ = Ph1B1
14 (1) d0z = rPe0 = il = jk
17 (1) P 2u
20 (1) d0P3e0 = e0P
3d0 = Pd0P2e0 = Pe0P
2d0
21 (1) h0d0P3e0 = h0e0P
3d0 = h0Pd0P2e0 = h0Pe0P
2d0 = h2d0P3d0 =
h2Pd0P2d0 = d2
0P3h2 = d0Ph2P
2d0 = d0Pd0P2h2 = gP 4h2 =
Ph2Pd20
22 (1) h20d0P
3e0 = h20e0P
3d0 = h20Pd0P
2e0 = h20Pe0P
2d0 =h0h2d0P
3d0 = h0h2Pd0P2d0 = h0d
20P
3h2 = h0d0Ph2P2d0 =
h0d0Pd0P2h2 = h0gP 4h2 = h0Ph2Pd2
0 = h22P
4e0 = h2e0P4h2 =
h2Ph2P3e0 = h2Pe0P
3h2 = h2P2h2P
2e0 = e0Ph2P3h2 =
e0P2h2
2 = Ph22P
2e0 = Ph2Pe0P2h2 = Pc0P
2i = iP 3c0
23 (1) P 4i
24 (1) h0P4i
25 (01) h20P
4i
(11) h1P5d0 = d0P
5h1 = Ph1P4d0 = Pd0P
4h1 = P 2h1P3d0 =
P 2d0P3h1
26 (1) h30P
4i
27 (1) h40P
4i
28 (1) h50P
4i
34 ROBERT R. BRUNER
Stem 569 (1) h5Pe0
10 (10) gt
(01) Q1
11 (1) h0Q1 = h2x′
12 (1) h20Q1 = h0h2x
′
13 (10) d0v
(01) h30Q1 = h2
0h2x′ = h3
1x′ = h1Ph1B1
(11) e0u = rj
16 (1) d40 = d0e0Pe0 = d0gPd0 = e2
0Pd0
18 (1) h1P2u = Ph1Q = Ph1Pu = P 2h1u
19 (1) d0P2j = e0P
2i = iP 2e0 = jP 2d0 = Pd0Pj
20 (1) h0d0P2j = h0e0P
2i = h0iP2e0 = h0jP
2d0 = h0Pd0Pj =h2d0P
2i = h2iP2d0 = d0iP
2h2 = f0P3d0 = Ph2iPd0 = kP 3h2
21 (1) h20d0P
2j = h20e0P
2i = h20iP
2e0 = h20jP
2d0 = h20Pd0Pj =
h0h2d0P2i = h0h2iP
2d0 = h0d0iP2h2 = h0f0P
3d0 =h0Ph2iPd0 = h0kP 3h2 = h1d0P
3e0 = h1e0P3d0 = h1Pd0P
2e0 =h1Pe0P
2d0 = h22P
3j = h2Ph2P2j = h2jP
3h2 = h2P2h2Pj =
d0e0P3h1 = d0Ph1P
2e0 = d0Pe0P2h1 = e0Ph1P
2d0 =e0Pd0P
2h1 = Ph1Pd0Pe0 = Ph22Pj = Ph2jP
2h2
26 (1) h21P
5d0 = h1d0P5h1 = h1Ph1P
4d0 = h1Pd0P4h1 =
h1P2h1P
3d0 = h1P2d0P
3h1 = h3P6h1 = c0P
5c0 = d0Ph1P4h1 =
d0P2h1P
3h1 = Ph21P
3d0 = Ph1Pd0P3h1 = Ph1P
2h1P2d0 =
Pc0P4c0 = Pd0P
2h21 = P 2c0P
3c0
27 (1) P 6c0
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 35
Stem 577 (1) Q2
8 (10) h5j
(01) h0Q2
9 (10) h0h5j = h2h5i
(01) h20Q2
10 (1) h20h5j = h0h2h5i = h1h5Pe0 = h4v = h5e0Ph1
11 (1) h1Q1 = h2R1
12 (1) e0g2
15 (1) d20k = d0e0j = d0gi = e2
0i = lP e0 = mPd0
18 (1) iP j
19 (1) h0iP j = h21P
2u = h1Ph1Q = h1Ph1Pu = h1P2h1u = f0P
2i =Ph2
1u = Ph2i2 = rP 3h2 = qP 3h1
24 (1) P 5e0
25 (1) h0P5e0 = h2P
5d0 = d0P5h2 = Ph2P
4d0 = Pd0P4h2 =
P 2h2P3d0 = P 2d0P
3h2
26 (1) h20P
5e0 = h0h2P5d0 = h0d0P
5h2 = h0Ph2P4d0 = h0Pd0P
4h2 =h0P
2h2P3d0 = h0P
2d0P3h2
27 (1) h30P
5e0 = h20h2P
5d0 = h20d0P
5h2 = h20Ph2P
4d0 = h20Pd0P
4h2 =h2
0P2h2P
3d0 = h20P
2d0P3h2 = h3
1P5d0 = h2
1d0P5h1 =
h21Ph1P
4d0 = h21Pd0P
4h1 = h21P
2h1P3d0 = h2
1P2d0P
3h1 =h1h3P
6h1 = h1c0P5c0 = h1d0Ph1P
4h1 = h1d0P2h1P
3h1 =h1Ph2
1P3d0 = h1Ph1Pd0P
3h1 = h1Ph1P2h1P
2d0 =h1Pc0P
4c0 = h1Pd0P2h2
1 = h1P2c0P
3c0 = h22P
6h2 =h2Ph2P
5h2 = h2P2h2P
4h2 = h2P3h2
2 = h3Ph1P5h1 =
h3P2h1P
4h1 = h3P3h2
1 = c20P
5h1 = c0Ph1P4c0 = c0Pc0P
4h1 =c0P
2h1P3c0 = c0P
2c0P3h1 = d0Ph2
1P3h1 = d0Ph1P
2h21 =
Ph31P
2d0 = Ph21Pd0P
2h1 = Ph1Pc0P3c0 = Ph1P
2c20 =
Ph22P
4h2 = Ph2P2h2P
3h2 = Pc20P
3h1 = Pc0P2h1P
2c0 = P 2h32
28 (1) h1P6c0 = c0P
6h1 = Ph1P5c0 = Pc0P
5h1 = P 2h1P4c0 =
P 2c0P4h1 = P 3h1P
3c0
29 (1) P 7h1
36 ROBERT R. BRUNER
Stem 586 (1) D2
7 (1) h0D2
8 (1) h20D2 = h1Q2
14 (1) d20r = e0z = im = jl = k2
17 (1) P 2v
20 (1) d20P
2d0 = d0Pd20 = e0P
3e0 = gP 3d0 = Pe0P2e0
21 (1) h0d20P
2d0 = h0d0Pd20 = h0e0P
3e0 = h0gP 3d0 = h0Pe0P2e0 =
h2d0P3e0 = h2e0P
3d0 = h2Pd0P2e0 = h2Pe0P
2d0 = d0e0P3h2 =
d0Ph2P2e0 = d0Pe0P
2h2 = e0Ph2P2d0 = e0Pd0P
2h2 =Ph2Pd0Pe0
22 (1) h20d
20P
2d0 = h20d0Pd2
0 = h20e0P
3e0 = h20gP 3d0 = h2
0Pe0P2e0 =
h0h2d0P3e0 = h0h2e0P
3d0 = h0h2Pd0P2e0 = h0h2Pe0P
2d0 =h0d0e0P
3h2 = h0d0Ph2P2e0 = h0d0Pe0P
2h2 = h0e0Ph2P2d0 =
h0e0Pd0P2h2 = h0Ph2Pd0Pe0 = h2
2d0P3d0 = h2
2Pd0P2d0 =
h2d20P
3h2 = h2d0Ph2P2d0 = h2d0Pd0P
2h2 = h2gP 4h2 =h2Ph2Pd2
0 = h4P5h2 = c0P
3j = d20Ph2P
2h2 = d0Ph22Pd0 =
gPh2P3h2 = gP 2h2
2 = Pc0P2j = jP 3c0 = P 2c0Pj
23 (1) P 4j
24 (1) h0P4j = h2P
4i = iP 4h2 = P 2h2P2i
25 (1) h20P
4j = h0h2P4i = h0iP
4h2 = h0P2h2P
2i = h1P5e0 = e0P
5h1 =Ph1P
4e0 = Pe0P4h1 = P 2h1P
3e0 = P 2e0P3h1
30 (1) h1P7h1 = Ph1P
6h1 = P 2h1P5h1 = P 3h1P
4h1
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 37
Stem 5910 (1) B21
11 (1) h0B21 = h2Q1
12 (1) h20B21 = h0h2Q1 = h2
2x′ = Ph2B2
13 (1) d0w = e0v = gu = rk
16 (1) d30e0 = d0gPe0 = e2
0Pe0 = e0gPd0
19 (1) d0iPd0 = e0P2j = gP 2i = jP 2e0 = kP 2d0 = Pe0Pj
20 (1) h0d0iPd0 = h0e0P2j = h0gP 2i = h0jP
2e0 = h0kP 2d0 =h0Pe0Pj = h2d0P
2j = h2e0P2i = h2iP
2e0 = h2jP2d0 =
h2Pd0Pj = d0Ph2Pj = d0jP2h2 = e0iP
2h2 = f0P3e0 =
Ph2iPe0 = Ph2jPd0 = lP 3h2
21 (1) h20d0iPd0 = h2
0e0P2j = h2
0gP 2i = h20jP
2e0 = h20kP 2d0 =
h20Pe0Pj = h0h2d0P
2j = h0h2e0P2i = h0h2iP
2e0 = h0h2jP2d0 =
h0h2Pd0Pj = h0d0Ph2Pj = h0d0jP2h2 = h0e0iP
2h2 =h0f0P
3e0 = h0Ph2iPe0 = h0Ph2jPd0 = h0lP3h2 = h1d
20P
2d0 =h1d0Pd2
0 = h1e0P3e0 = h1gP 3d0 = h1Pe0P
2e0 = h22d0P
2i =h2
2iP2d0 = h2d0iP
2h2 = h2f0P3d0 = h2Ph2iPd0 = h2kP 3h2 =
d30P
2h1 = d20Ph1Pd0 = d0f0P
3h2 = d0gP 3h1 = d0Ph22i =
e20P
3h1 = e0Ph1P2e0 = e0Pe0P
2h1 = f0Ph2P2d0 =
f0Pd0P2h2 = gPh1P
2d0 = gPd0P2h1 = Ph1Pe2
0 = Ph2kP 2h2
29 (1) P 7h2
30 (1) h0P7h2
31 (1) h20P
7h2 = h21P
7h1 = h1Ph1P6h1 = h1P
2h1P5h1 =
h1P3h1P
4h1 = Ph21P
5h1 = Ph1P2h1P
4h1 = Ph1P3h2
1 =P 2h2
1P3h1
38 ROBERT R. BRUNER
Stem 607 (1) B3
8 (1) h0B3 = h2Q2 = h5k
9 (10) B4
(01) h20B3 = h0h2Q2 = h0h5k = h2h5j
10 (1) h0B4
11 (10) h3x′
(01) h20B4
(11) h1B21 = d0B1
12 (10) g3
(01) h30B4 = h0h3x
′ = xi
(11) r2
13 (1) h40B4 = h2
0h3x′ = h0xi = h0r
2
14 (1) h50B4 = h3
0h3x′ = h2
0xi = h20r
2 = h3ri
15 (1) d20l = d0e0k = d0gj = e2
0j = e0gi = mPe0
18 (1) d0i2 = rP 2d0 = jP j
24 (1) d0P4d0 = Pd0P
3d0 = P 2d20
25 (1) h0d0P4d0 = h0Pd0P
3d0 = h0P2d2
0 = h2P5e0 = e0P
5h2 =Ph2P
4e0 = Pe0P4h2 = P 2h2P
3e0 = P 2e0P3h2
26 (1) h20d0P
4d0 = h20Pd0P
3d0 = h20P
2d20 = h0h2P
5e0 = h0e0P5h2 =
h0Ph2P4e0 = h0Pe0P
4h2 = h0P2h2P
3e0 = h0P2e0P
3h2 =h2
2P5d0 = h2d0P
5h2 = h2Ph2P4d0 = h2Pd0P
4h2 =h2P
2h2P3d0 = h2P
2d0P3h2 = d0Ph2P
4h2 = d0P2h2P
3h2 =Ph2
2P3d0 = Ph2Pd0P
3h2 = Ph2P2h2P
2d0 = Pd0P2h2
2
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 39
Stem 614 (1) D3
6 (01) A
(11) A′
7 (01) h0A = h2D2
(11) h0A′
8 (1) h20A = h0h2D2 = h1B3 = h4B1
9 (1) X1
10 (1) h0X1
11 (10) nr
(01) h20X1 = h3R1
12 (1) h30X1 = h0h3R1
13 (1) h40X1 = h2
0h3R1 = c0x′ = yi
14 (1) d0e0r = gz = jm = kl
17 (1) d0Pu = Pd0u
20 (1) d20P
2e0 = d0e0P2d0 = d0Pd0Pe0 = e0Pd2
0 = gP 3e0
23 (1) iP 3d0 = Pd0P2i
24 (1) h0iP3d0 = h0Pd0P
2i = h2P4j = Ph2P
3j = jP 4h2 = P 2h2P2j =
PjP 3h2
25 (1) h20iP
3d0 = h20Pd0P
2i = h0h2P4j = h0Ph2P
3j = h0jP4h2 =
h0P2h2P
2j = h0PjP 3h2 = h1d0P4d0 = h1Pd0P
3d0 = h1P2d2
0 =h2
2P4i = h2iP
4h2 = h2P2h2P
2i = d20P
4h1 = d0Ph1P3d0 =
d0Pd0P3h1 = d0P
2h1P2d0 = f0P
5h2 = gP 5h1 = Ph1Pd0P2d0 =
Ph22P
2i = Ph2iP3h2 = iP 2h2
2 = Pd20P
2h1
40 ROBERT R. BRUNER
Stem 622 (1) h2
5
3 (1) h0h25
4 (1) h20h
25
5 (100) H1
(010) h1D3
(001) h30h
25
6 (10) h5n
(01) h40h
25
7 (1) h50h
25
8 (100) x8,32
(010) x8,33
(001) h60h
25
9 (1) h70h
25 = h0x8,32 = h0x8,33
10 (100) x10,27
(010) x10,28
(001) h1X1
11 (10) h0x10,27
(01) h0x10,28 = h2B21 = d0B2
12 (10) h20x10,27
(01) h20x10,28 = h0h2B21 = h0d0B2 = h2
2Q1
13 (10) e0w = gv = rl
(01) h30x10,27
14 (1) h40x10,27 = h4Q
15 (10) Ph1x′
(01) h50x10,27 = h0h4Q
16 (10) d30g = d2
0e20 = e0gPe0 = g2Pd0 = iu
(01) h60x10,27 = h2
0h4Q
17 (1) h70x10,27 = h3
0h4Q
18 (1) h80x10,27 = h4
0h4Q
19 (01) h90x10,27 = h5
0h4Q
(11) d20Pj = d0iPe0 = d0jPd0 = e0iPd0 = gP 2j = kP 2e0 = lP 2d0
20 (1) h100 x10,27 = h6
0h4Q = h0d20Pj = h0d0iPe0 = h0d0jPd0 =
h0e0iPd0 = h0gP 2j = h0kP 2e0 = h0lP2d0 = h2d0iPd0 =
h2e0P2j = h2gP 2i = h2jP
2e0 = h2kP 2d0 = h2Pe0Pj =d20Ph2i = d0f0P
2d0 = d0kP 2h2 = e0Ph2Pj = e0jP2h2 =
f0Pd20 = giP 2h2 = Ph2jPe0 = Ph2kPd0 = mP 3h2
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 41
Stem 62 continued21 (1) h11
0 x10,27 = h70h4Q = h2
0d20Pj = h2
0d0iPe0 = h20d0jPd0 =
h20e0iPd0 = h2
0gP 2j = h20kP 2e0 = h2
0lP2d0 = h0h2d0iPd0 =
h0h2e0P2j = h0h2gP 2i = h0h2jP
2e0 = h0h2kP 2d0 =h0h2Pe0Pj = h0d
20Ph2i = h0d0f0P
2d0 = h0d0kP 2h2 =h0e0Ph2Pj = h0e0jP
2h2 = h0f0Pd20 = h0giP 2h2 = h0Ph2jPe0 =
h0Ph2kPd0 = h0mP 3h2 = h1d20P
2e0 = h1d0e0P2d0 =
h1d0Pd0Pe0 = h1e0Pd20 = h1gP 3e0 = h2
2d0P2j = h2
2e0P2i =
h22iP
2e0 = h22jP
2d0 = h22Pd0Pj = h2d0Ph2Pj = h2d0jP
2h2 =h2e0iP
2h2 = h2f0P3e0 = h2Ph2iPe0 = h2Ph2jPd0 = h2lP
3h2 =d20e0P
2h1 = d20Ph1Pe0 = d0e0Ph1Pd0 = d0Ph2
2j = e0f0P3h2 =
e0gP 3h1 = e0Ph22i = f0Ph2P
2e0 = f0Pe0P2h2 = gPh1P
2e0 =gPe0P
2h1 = Ph2lP2h2 = rP 3c0 = Pc0i
2
22 (1) iP 2i
23 (1) h0iP2i
24 (1) h20iP
2i = h3P4i
25 (1) h30iP
2i = h0h3P4i
26 (1) h40iP
2i = h20h3P
4i
27 (1) h50iP
2i = h30h3P
4i = c0P5d0 = d0P
5c0 = Pc0P4d0 = Pd0P
4c0 =P 2c0P
3d0 = P 2d0P3c0
28 (1) P 6d0
29 (1) h0P6d0
30 (1) h20P
6d0 = h2P7h2 = Ph2P
6h2 = P 2h2P5h2 = P 3h2P
4h2
42 ROBERT R. BRUNER
Stem 631 (1) h6
2 (1) h0h6
3 (10) h1h25
(01) h20h6
4 (1) h30h6
5 (1) h40h6
6 (10) h1H1
(01) h50h6
7 (100) x7,33
(010) x7,34
(001) h60h6
8 (10) h0x7,33 = h0x7,34 = h2B3 = h4B2 = h5l
(01) h70h6
9 (100) h1x8,32
(010) h20x7,33 = h2
0x7,34 = h0h2B3 = h0h4B2 = h0h5l = h22Q2 =
h2h5k = h5d0f0
(001) h80h6
10 (10) h2B4
(01) h90h6
11 (100) h21X1 = h3Q1 = Ph1G
(010) h0h2B4 = h1x10,28 = e0B1
(110) h1x10,27
(001) h100 h6
12 (1) h110 h6
13 (1) h120 h6
14 (1) h130 h6
15 (10) d20m = d0e0l = d0gk = e2
0k = e0gj = g2i
(01) h140 h6
16 (10) h1Ph1x′ = B1P
2h1
(01) h150 h6
17 (1) h160 h6
18 (10) d0ij = e0i2 = rP 2e0 = kPj = Pd0z
(01) h170 h6
19 (1) h180 h6
20 (1) h190 h6
21 (10) P 3u
(01) h200 h6
22 (1) h210 h6
23 (1) h220 h6
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 43
Stem 63 continued24 (10) d0P
4e0 = e0P4d0 = Pd0P
3e0 = Pe0P3d0 = P 2d0P
2e0
(01) h230 h6
25 (10) h0d0P4e0 = h0e0P
4d0 = h0Pd0P3e0 = h0Pe0P
3d0 =h0P
2d0P2e0 = h2d0P
4d0 = h2Pd0P3d0 = h2P
2d20 = d2
0P4h2 =
d0Ph2P3d0 = d0Pd0P
3h2 = d0P2h2P
2d0 = gP 5h2 =Ph2Pd0P
2d0 = Pd20P
2h2
(01) h240 h6
26 (10) h20d0P
4e0 = h20e0P
4d0 = h20Pd0P
3e0 = h20Pe0P
3d0 =h2
0P2d0P
2e0 = h0h2d0P4d0 = h0h2Pd0P
3d0 = h0h2P2d2
0 =h0d
20P
4h2 = h0d0Ph2P3d0 = h0d0Pd0P
3h2 = h0d0P2h2P
2d0 =h0gP 5h2 = h0Ph2Pd0P
2d0 = h0Pd20P
2h2 = h22P
5e0 =h2e0P
5h2 = h2Ph2P4e0 = h2Pe0P
4h2 = h2P2h2P
3e0 =h2P
2e0P3h2 = c0P
4i = e0Ph2P4h2 = e0P
2h2P3h2 = Ph2
2P3e0 =
Ph2Pe0P3h2 = Ph2P
2h2P2e0 = iP 4c0 = Pe0P
2h22 = P 2c0P
2i
(01) h250 h6
27 (1) h260 h6
28 (1) h270 h6
29 (10) h1P6d0 = d0P
6h1 = Ph1P5d0 = Pd0P
5h1 = P 2h1P4d0 =
P 2d0P4h1 = P 3h1P
3d0
(01) h280 h6
30 (1) h290 h6
31 (1) h300 h6
32 (1) h310 h6
44 ROBERT R. BRUNER
Stem 642 (1) h1h6
4 (1) h21h
25
5 (1) h2D3
6 (1) A′′
7 (10) h2A
(01) h0A′′
(11) h2A′
8 (100) h3Q2
(010) h0h2A = h1x7,34 = h22D2
(001) h20A
′′ = h1x7,33
(011) h0h2A′ = gg2 = d2
1
9 (1) h0h3Q2
10 (10) x10,32
(01) h20h3Q2 = h2
1x8,32 = d1q
14 (10) PQ1
(01) d0gr = e20r = km = l2
15 (1) h0PQ1 = Ph2x′
16 (1) h20PQ1 = h0Ph2x
′
17 (10) d0Pv = Pd0v
(01) h30PQ1 = h2
0Ph2x′ = h2
1Ph1x′ = h1B1P
2h1 = e0Q = Ph21B1
(11) e0Pu = rPj = iz = Pe0u
20 (1) d30Pd0 = d0e0P
2e0 = d0gP 2d0 = d0Pe20 = e2
0P2d0 = e0Pd0Pe0 =
gPd20
22 (1) h1P3u = Ph1P
2u = P 2h1Q = P 2h1Pu = uP 3h1
23 (1) d0P3j = iP 3e0 = jP 3d0 = Pd0P
2j = Pe0P2i = PjP 2d0
24 (1) h0d0P3j = h0iP
3e0 = h0jP3d0 = h0Pd0P
2j = h0Pe0P2i =
h0PjP 2d0 = h2iP3d0 = h2Pd0P
2i = d0Ph2P2i = d0iP
3h2 =f0P
4d0 = Ph2iP2d0 = iPd0P
2h2 = kP 4h2
25 (1) h20d0P
3j = h20iP
3e0 = h20jP
3d0 = h20Pd0P
2j = h20Pe0P
2i =h2
0PjP 2d0 = h0h2iP3d0 = h0h2Pd0P
2i = h0d0Ph2P2i =
h0d0iP3h2 = h0f0P
4d0 = h0Ph2iP2d0 = h0iPd0P
2h2 =h0kP 4h2 = h1d0P
4e0 = h1e0P4d0 = h1Pd0P
3e0 = h1Pe0P3d0 =
h1P2d0P
2e0 = h22P
4j = h2Ph2P3j = h2jP
4h2 = h2P2h2P
2j =h2PjP 3h2 = d0e0P
4h1 = d0Ph1P3e0 = d0Pe0P
3h1 =d0P
2h1P2e0 = e0Ph1P
3d0 = e0Pd0P3h1 = e0P
2h1P2d0 =
Ph1Pd0P2e0 = Ph1Pe0P
2d0 = Ph22P
2j = Ph2jP3h2 =
Ph2P2h2Pj = jP 2h2
2 = Pd0Pe0P2h1
30 (1) h21P
6d0 = h1d0P6h1 = h1Ph1P
5d0 = h1Pd0P5h1 =
h1P2h1P
4d0 = h1P2d0P
4h1 = h1P3h1P
3d0 = h3P7h1 =
c0P6c0 = d0Ph1P
5h1 = d0P2h1P
4h1 = d0P3h2
1 = Ph21P
4d0 =Ph1Pd0P
4h1 = Ph1P2h1P
3d0 = Ph1P2d0P
3h1 = Pc0P5c0 =
Pd0P2h1P
3h1 = P 2h21P
2d0 = P 2c0P4c0 = P 3c2
0
31 (1) P 7c0
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 45
Stem 653 (10) h2h
25
(01) h21h6
4 (1) h0h2h25
5 (1) h20h2h
25 = h3
1h25
6 (1) h2H1
7 (10) h2h5n = h4C
(01) h3D2
8 (1) h0h3D2
9 (10) h2x8,33
(01) h20h3D2 = h1h3Q2
(11) h2x8,32
10 (1) B23
11 (10) h1x10,32
(01) h0B23 = h2x10,27 = h2x10,28 = e0B2
12 (10) h5Pj
(01) h20B23 = h0h2x10,27 = h0h2x10,28 = h0e0B2 = h2
2B21 = h2d0B2
13 (10) gw = rm
(01) R2
14 (1) h0R2
15 (1) h20R2 = h1PQ1 = Ph1Q1 = Ph2R1
16 (1) d20e0g = d0e
30 = g2Pe0 = iv = ju
19 (1) d30i = d0e0Pj = d0jPe0 = d0kPd0 = e0iPe0 = e0jPd0 = giPd0 =
lP 2e0 = mP 2d0
22 (1) iP 2j = jP 2i
23 (1) h0iP2j = h0jP
2i = h21P
3u = h1Ph1P2u = h1P
2h1Q =h1P
2h1Pu = h1uP 3h1 = h2iP2i = Ph2
1Q = Ph21Pu =
Ph1P2h1u = rP 4h2 = qP 4h1 = i2P 2h2
28 (1) P 6e0
29 (1) h0P6e0 = h2P
6d0 = d0P6h2 = Ph2P
5d0 = Pd0P5h2 =
P 2h2P4d0 = P 2d0P
4h2 = P 3h2P3d0
30 (1) h20P
6e0 = h0h2P6d0 = h0d0P
6h2 = h0Ph2P5d0 = h0Pd0P
5h2 =h0P
2h2P4d0 = h0P
2d0P4h2 = h0P
3h2P3d0
continued
46 ROBERT R. BRUNER
Stem 65 continued31 (1) h3
0P6e0 = h2
0h2P6d0 = h2
0d0P6h2 = h2
0Ph2P5d0 = h2
0Pd0P5h2 =
h20P
2h2P4d0 = h2
0P2d0P
4h2 = h20P
3h2P3d0 = h3
1P6d0 =
h21d0P
6h1 = h21Ph1P
5d0 = h21Pd0P
5h1 = h21P
2h1P4d0 =
h21P
2d0P4h1 = h2
1P3h1P
3d0 = h1h3P7h1 = h1c0P
6c0 =h1d0Ph1P
5h1 = h1d0P2h1P
4h1 = h1d0P3h2
1 = h1Ph21P
4d0 =h1Ph1Pd0P
4h1 = h1Ph1P2h1P
3d0 = h1Ph1P2d0P
3h1 =h1Pc0P
5c0 = h1Pd0P2h1P
3h1 = h1P2h2
1P2d0 = h1P
2c0P4c0 =
h1P3c2
0 = h22P
7h2 = h2Ph2P6h2 = h2P
2h2P5h2 =
h2P3h2P
4h2 = h3Ph1P6h1 = h3P
2h1P5h1 = h3P
3h1P4h1 =
c20P
6h1 = c0Ph1P5c0 = c0Pc0P
5h1 = c0P2h1P
4c0 =c0P
2c0P4h1 = c0P
3h1P3c0 = d0Ph2
1P4h1 = d0Ph1P
2h1P3h1 =
d0P2h3
1 = Ph31P
3d0 = Ph21Pd0P
3h1 = Ph21P
2h1P2d0 =
Ph1Pc0P4c0 = Ph1Pd0P
2h21 = Ph1P
2c0P3c0 = Ph2
2P5h2 =
Ph2P2h2P
4h2 = Ph2P3h2
2 = Pc20P
4h1 = Pc0P2h1P
3c0 =Pc0P
2c0P3h1 = P 2h1P
2c20 = P 2h2
2P3h2
32 (1) h1P7c0 = c0P
7h1 = Ph1P6c0 = Pc0P
6h1 = P 2h1P5c0 =
P 2c0P5h1 = P 3h1P
4c0 = P 3c0P4h1
33 (1) P 8h1
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 47
Stem 662 (1) h2h6
3 (1) h0h2h6
4 (1) h20h2h6 = h3
1h6
6 (1) r1
7 (10) x7,40
(01) h0r1
8 (1) h0x7,40 = h2x7,34
10 (10) B5
(01) PD2
11 (01) h0PD2
(11) h0B5 = h1B23 = h22B4 = f0B2 = gB1
12 (10) gN = nm = rt
(01) h20PD2 = h2
1x10,32 = Ph1Q2
15 (1) d0e0m = d0gl = e20l = e0gk = g2j
18 (1) d0rPd0 = d0ik = d0j2 = e0ij = gi2 = lP j = Pe0z
21 (1) P 3v
24 (1) d20P
3d0 = d0Pd0P2d0 = e0P
4e0 = gP 4d0 = Pd30 = Pe0P
3e0 =P 2e2
0
25 (1) h0d20P
3d0 = h0d0Pd0P2d0 = h0e0P
4e0 = h0gP 4d0 = h0Pd30 =
h0Pe0P3e0 = h0P
2e20 = h2d0P
4e0 = h2e0P4d0 = h2Pd0P
3e0 =h2Pe0P
3d0 = h2P2d0P
2e0 = d0e0P4h2 = d0Ph2P
3e0 =d0Pe0P
3h2 = d0P2h2P
2e0 = e0Ph2P3d0 = e0Pd0P
3h2 =e0P
2h2P2d0 = Ph2Pd0P
2e0 = Ph2Pe0P2d0 = Pd0Pe0P
2h2
26 (1) h20d
20P
3d0 = h20d0Pd0P
2d0 = h20e0P
4e0 = h20gP 4d0 = h2
0Pd30 =
h20Pe0P
3e0 = h20P
2e20 = h0h2d0P
4e0 = h0h2e0P4d0 =
h0h2Pd0P3e0 = h0h2Pe0P
3d0 = h0h2P2d0P
2e0 =h0d0e0P
4h2 = h0d0Ph2P3e0 = h0d0Pe0P
3h2 = h0d0P2h2P
2e0 =h0e0Ph2P
3d0 = h0e0Pd0P3h2 = h0e0P
2h2P2d0 =
h0Ph2Pd0P2e0 = h0Ph2Pe0P
2d0 = h0Pd0Pe0P2h2 =
h22d0P
4d0 = h22Pd0P
3d0 = h22P
2d20 = h2d
20P
4h2 =h2d0Ph2P
3d0 = h2d0Pd0P3h2 = h2d0P
2h2P2d0 = h2gP 5h2 =
h2Ph2Pd0P2d0 = h2Pd2
0P2h2 = h4P
6h2 = c0P4j =
d20Ph2P
3h2 = d20P
2h22 = d0Ph2
2P2d0 = d0Ph2Pd0P
2h2 =gPh2P
4h2 = gP 2h2P3h2 = Ph2
2Pd20 = Pc0P
3j = jP 4c0 =P 2c0P
2j = PjP 3c0
27 (1) P 5j
28 (1) h0P5j = Ph2P
4i = iP 5h2 = P 3h2P2i
29 (1) h20P
5j = h0Ph2P4i = h0iP
5h2 = h0P3h2P
2i = h1P6e0 =
e0P6h1 = Ph1P
5e0 = Pe0P5h1 = P 2h1P
4e0 = P 2e0P4h1 =
P 3h1P3e0
34 (1) h1P8h1 = Ph1P
7h1 = P 2h1P6h1 = P 3h1P
5h1 = P 4h21
48 ROBERT R. BRUNER
Stem 675 (10) n1
(01) Q3
6 (01) h0Q3
(11) h0n1 = h22D3 = h4D1
7 (1) h20Q3 = h2A
′′
8 (1) h1x7,40 = h22A
′ = h22A
9 (10) x9,39
(01) x9,40
10 (1) h0x9,40 = h3B4
11 (10) x11,35
(01) h20x9,40 = h0h3B4 = g2i = xr
12 (1) h30x9,40 = h2
0h3B4 = h0g2i = h0xr = h0x11,35 = h23x
′
14 (10) e0gr = lm
(01) d0x′
15 (1) h0d0x′ = h2PQ1 = Ph2Q1
16 (1) h20d0x
′ = h0h2PQ1 = h0Ph2Q1 = h2Ph2x′ = B2P
2h2
17 (1) d20u = d0ri = e0Pv = gPu = jz = Pd0w = Pe0v
20 (1) d30Pe0 = d2
0e0Pd0 = d0gP 2e0 = e20P
2e0 = e0gP 2d0 = e0Pe20 =
gPd0Pe0
23 (1) d20P
2i = d0iP2d0 = e0P
3j = iPd20 = jP 3e0 = kP 3d0 = Pe0P
2j =PjP 2e0
24 (1) h0d20P
2i = h0d0iP2d0 = h0e0P
3j = h0iPd20 = h0jP
3e0 =h0kP 3d0 = h0Pe0P
2j = h0PjP 2e0 = h2d0P3j = h2iP
3e0 =h2jP
3d0 = h2Pd0P2j = h2Pe0P
2i = h2PjP 2d0 = d0Ph2P2j =
d0jP3h2 = d0P
2h2Pj = e0Ph2P2i = e0iP
3h2 = f0P4e0 =
Ph2iP2e0 = Ph2jP
2d0 = Ph2Pd0Pj = iPe0P2h2 = jPd0P
2h2 =lP 4h2
25 (1) h20d
20P
2i = h20d0iP
2d0 = h20e0P
3j = h20iPd2
0 = h20jP
3e0 =h2
0kP 3d0 = h20Pe0P
2j = h20PjP 2e0 = h0h2d0P
3j = h0h2iP3e0 =
h0h2jP3d0 = h0h2Pd0P
2j = h0h2Pe0P2i = h0h2PjP 2d0 =
h0d0Ph2P2j = h0d0jP
3h2 = h0d0P2h2Pj = h0e0Ph2P
2i =h0e0iP
3h2 = h0f0P4e0 = h0Ph2iP
2e0 = h0Ph2jP2d0 =
h0Ph2Pd0Pj = h0iPe0P2h2 = h0jPd0P
2h2 = h0lP4h2 =
h1d20P
3d0 = h1d0Pd0P2d0 = h1e0P
4e0 = h1gP 4d0 = h1Pd30 =
h1Pe0P3e0 = h1P
2e20 = h2
2iP3d0 = h2
2Pd0P2i = h2d0Ph2P
2i =h2d0iP
3h2 = h2f0P4d0 = h2Ph2iP
2d0 = h2iPd0P2h2 =
h2kP 4h2 = d30P
3h1 = d20Ph1P
2d0 = d20Pd0P
2h1 = d0f0P4h2 =
d0gP 4h1 = d0Ph1Pd20 = d0Ph2iP
2h2 = e20P
4h1 = e0Ph1P3e0 =
e0Pe0P3h1 = e0P
2h1P2e0 = f0Ph2P
3d0 = f0Pd0P3h2 =
f0P2h2P
2d0 = gPh1P3d0 = gPd0P
3h1 = gP 2h1P2d0 =
Ph1Pe0P2e0 = Ph2
2iPd0 = Ph2kP 3h2 = kP 2h22 = Pe2
0P2h1
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 49
Stem 67 continued33 (1) P 8h2
34 (1) h0P8h2
35 (1) h20P
8h2 = h21P
8h1 = h1Ph1P7h1 = h1P
2h1P6h1 =
h1P3h1P
5h1 = h1P4h2
1 = Ph21P
6h1 = Ph1P2h1P
5h1 =Ph1P
3h1P4h1 = P 2h2
1P4h1 = P 2h1P
3h21
Stem 684 (1) d2
5 (1) h0d2 = h3D3
6 (1) h1Q3
7 (10) h22H1 = h3A
(11) h3A′
8 (10) G21
(01) h0h3A′
9 (1) h0G21
10 (10) h1x9,39 = h22x8,32 = h2
2x8,33 = f0C = d1t = e1r
(01) h20G21
(11) h3X1
11 (10) h2B23 = gB2
(01) h30G21 = h0h3X1
(11) h4x′
12 (10) h0h4x′ = h5d0i
(01) h40G21 = h2
0h3X1 = h23R1 = ry
(11) h0h2B23 = h0gB2 = h22x10,27 = h2
2x10,28 = h2e0B2 = c0B4 =Ph2Q2
13 (10) P 2D1
(01) h50G21 = h3
0h3X1 = h20h4x
′ = h0h23R1 = h0h5d0i = h0ry =
h2h5Pj = h5Ph2j
14 (1) h0P2D1 = h2R2 = d0R1
15 (1) h20P
2D1 = h0h2R2 = h0d0R1 = h1d0x′ = Ph1B21 = B1Pd0
16 (1) d20g
2 = d0e20g = e4
0 = iw = jv = ku
19 (1) d30j = d2
0e0i = d0gPj = d0kPe0 = d0lPd0 = e20Pj = e0jPe0 =
e0kPd0 = giPe0 = gjPd0 = mP 2e0
22 (1) rP 3d0 = i2Pd0 = jP 2j = kP 2i = Pj2
28 (1) d0P5d0 = Pd0P
4d0 = P 2d0P3d0
29 (1) h0d0P5d0 = h0Pd0P
4d0 = h0P2d0P
3d0 = h2P6e0 = e0P
6h2 =Ph2P
5e0 = Pe0P5h2 = P 2h2P
4e0 = P 2e0P4h2 = P 3h2P
3e0
30 (1) h20d0P
5d0 = h20Pd0P
4d0 = h20P
2d0P3d0 = h0h2P
6e0 =h0e0P
6h2 = h0Ph2P5e0 = h0Pe0P
5h2 = h0P2h2P
4e0 =h0P
2e0P4h2 = h0P
3h2P3e0 = h2
2P6d0 = h2d0P
6h2 =h2Ph2P
5d0 = h2Pd0P5h2 = h2P
2h2P4d0 = h2P
2d0P4h2 =
h2P3h2P
3d0 = d0Ph2P5h2 = d0P
2h2P4h2 = d0P
3h22 =
Ph22P
4d0 = Ph2Pd0P4h2 = Ph2P
2h2P3d0 = Ph2P
2d0P3h2 =
Pd0P2h2P
3h2 = P 2h22P
2d0
50 ROBERT R. BRUNER
Stem 693 (1) h2
2h6
4 (1) p′
5 (1) h0p′
6 (10) h3H1
(01) h20p
′
7 (1) h30p
′ = h21Q3 = h4G = c0D3
8 (10) h2x7,40
(01) PD3
9 (10) h1G21
(01) h0h2x7,40 = h22x7,34 = h3x8,33 = c0A
′ = c0A = c1C = e0D1 = e1n
(11) h3x8,32
10 (1) PA
11 (10) h2B5 = h4R1
(01) h0PA = h2PD2 = Ph2D2
(11) h3x10,27
13 (1) W1
15 (1) d0gm = e20m = e0gl = g2k = ru
18 (10) x18,20
(01) d20z = d0rPe0 = d0il = d0jk = e0rPd0 = e0ik = e0j
2 = gij =mPj
19 (1) h0x18,20
20 (1) h20x18,20
21 (10) d0P2u = Pd0Pu = uP 2d0
(01) h30x18,20
(11) rP 2i = i3
22 (1) h40x18,20 = h0rP
2i = h0i3
23 (1) h50x18,20 = h2
0rP2i = h2
0i3 = h3iP
2i
24 (01) h60x18,20 = h3
0rP2i = h3
0i3 = h0h3iP
2i
(11) d20P
3e0 = d0e0P3d0 = d0Pd0P
2e0 = d0Pe0P2d0 = e0Pd0P
2d0 =gP 4e0 = Pd2
0Pe0
25 (1) h70x18,20 = h4
0rP2i = h4
0i3 = h2
0h3iP2i = h0d
20P
3e0 =h0d0e0P
3d0 = h0d0Pd0P2e0 = h0d0Pe0P
2d0 = h0e0Pd0P2d0 =
h0gP 4e0 = h0Pd20Pe0 = h2d
20P
3d0 = h2d0Pd0P2d0 = h2e0P
4e0 =h2gP 4d0 = h2Pd3
0 = h2Pe0P3e0 = h2P
2e20 = h2
3P4i = h4P
5d0 =d30P
3h2 = d20Ph2P
2d0 = d20Pd0P
2h2 = d0gP 4h2 = d0Ph2Pd20 =
e20P
4h2 = e0Ph2P3e0 = e0Pe0P
3h2 = e0P2h2P
2e0 =gPh2P
3d0 = gPd0P3h2 = gP 2h2P
2d0 = Ph2Pe0P2e0 =
Pe20P
2h2
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 51
Stem 69 continued26 (1) h8
0x18,20 = h50rP
2i = h50i
3 = h30h3iP
2i = h20d
20P
3e0 =h2
0d0e0P3d0 = h2
0d0Pd0P2e0 = h2
0d0Pe0P2d0 = h2
0e0Pd0P2d0 =
h20gP 4e0 = h2
0Pd20Pe0 = h0h2d
20P
3d0 = h0h2d0Pd0P2d0 =
h0h2e0P4e0 = h0h2gP 4d0 = h0h2Pd3
0 = h0h2Pe0P3e0 =
h0h2P2e2
0 = h0h23P
4i = h0h4P5d0 = h0d
30P
3h2 = h0d20Ph2P
2d0 =h0d
20Pd0P
2h2 = h0d0gP 4h2 = h0d0Ph2Pd20 = h0e
20P
4h2 =h0e0Ph2P
3e0 = h0e0Pe0P3h2 = h0e0P
2h2P2e0 = h0gPh2P
3d0 =h0gPd0P
3h2 = h0gP 2h2P2d0 = h0Ph2Pe0P
2e0 = h0Pe20P
2h2 =h2
2d0P4e0 = h2
2e0P4d0 = h2
2Pd0P3e0 = h2
2Pe0P3d0 =
h22P
2d0P2e0 = h2d0e0P
4h2 = h2d0Ph2P3e0 = h2d0Pe0P
3h2 =h2d0P
2h2P2e0 = h2e0Ph2P
3d0 = h2e0Pd0P3h2 =
h2e0P2h2P
2d0 = h2Ph2Pd0P2e0 = h2Ph2Pe0P
2d0 =h2Pd0Pe0P
2h2 = c0iP3d0 = c0Pd0P
2i = d0e0Ph2P3h2 =
d0e0P2h2
2 = d0Ph22P
2e0 = d0Ph2Pe0P2h2 = d0Pc0P
2i =d0iP
3c0 = e0Ph22P
2d0 = e0Ph2Pd0P2h2 = Ph2
2Pd0Pe0 =Pc0iP
2d0 = iPd0P2c0 = kP 4c0
27 (1) d0P4i = iP 4d0 = P 2d0P
2i
28 (1) h0d0P4i = h0iP
4d0 = h0P2d0P
2i = h2P5j = Ph2P
4j = jP 5h2 =P 2h2P
3j = PjP 4h2 = P 3h2P2j
29 (1) h20d0P
4i = h20iP
4d0 = h20P
2d0P2i = h0h2P
5j = h0Ph2P4j =
h0jP5h2 = h0P
2h2P3j = h0PjP 4h2 = h0P
3h2P2j =
h1d0P5d0 = h1Pd0P
4d0 = h1P2d0P
3d0 = h2Ph2P4i =
h2iP5h2 = h2P
3h2P2i = d2
0P5h1 = d0Ph1P
4d0 = d0Pd0P4h1 =
d0P2h1P
3d0 = d0P2d0P
3h1 = f0P6h2 = gP 6h1 =
Ph1Pd0P3d0 = Ph1P
2d20 = Ph2iP
4h2 = Ph2P2h2P
2i =iP 2h2P
3h2 = Pd20P
3h1 = Pd0P2h1P
2d0
52 ROBERT R. BRUNER
Stem 702 (1) h3h6
3 (1) h0h3h6
4 (01) h20h3h6
(11) p1
5 (10) h1p′ = h2
5c0
(01) h30h3h6
6 (1) h2Q3
7 (10) h1h3H1
(01) h0h2Q3
8 (1) h3x7,33 = d1e1
9 (1) h1PD3 = D3Ph1
10 (1) h21G21 = h1h3x8,32 = h2x9,39 = gC = d1y = e1q
12 (1) h2x11,35
14 (010) d0Q1
(110) g2r = m2
(001) h1W1 = Ph1X1
(011) e0x′
15 (1) h0d0Q1 = h0e0x′ = h2d0x
′ = Ph2B21 = B2Pd0
16 (1) h20d0Q1 = h2
0e0x′ = h0h2d0x
′ = h0Ph2B21 = h0B2Pd0 =h2
2PQ1 = h2Ph2Q1
17 (10) R1
(01) d20v = d0e0u = d0rj = e0ri = gPv = kz = Pe0w
18 (1) h0R1
19 (10) h1x18,20 = P 2h1x′
(01) h20R1
20 (10) d50 = d2
0e0Pe0 = d20gPd0 = d0e
20Pd0 = e0gP 2e0 = g2P 2d0 =
gPe20 = iPu
(01) h30R1 = iQ
21 (1) h40R1 = h0iQ
22 (1) h50R1 = h2
0iQ
23 (01) h60R1 = h3
0iQ
(11) d20P
2j = d0e0P2i = d0iP
2e0 = d0jP2d0 = d0Pd0Pj = e0iP
2d0 =gP 3j = iPd0Pe0 = jPd2
0 = kP 3e0 = lP 3d0
24 (1) h70R1 = h4
0iQ = h0d20P
2j = h0d0e0P2i = h0d0iP
2e0 =h0d0jP
2d0 = h0d0Pd0Pj = h0e0iP2d0 = h0gP 3j = h0iPd0Pe0 =
h0jPd20 = h0kP 3e0 = h0lP
3d0 = h2d20P
2i = h2d0iP2d0 =
h2e0P3j = h2iPd2
0 = h2jP3e0 = h2kP 3d0 = h2Pe0P
2j =h2PjP 2e0 = d2
0iP2h2 = d0f0P
3d0 = d0Ph2iPd0 = d0kP 3h2 =e0Ph2P
2j = e0jP3h2 = e0P
2h2Pj = f0Pd0P2d0 = gPh2P
2i =giP 3h2 = Ph2jP
2e0 = Ph2kP 2d0 = Ph2Pe0Pj = jPe0P2h2 =
kPd0P2h2 = mP 4h2
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 53
Stem 70 continued25 (1) h8
0R1 = h50iQ = h2
0d20P
2j = h20d0e0P
2i = h20d0iP
2e0 =h2
0d0jP2d0 = h2
0d0Pd0Pj = h20e0iP
2d0 = h20gP 3j = h2
0iPd0Pe0 =h2
0jPd20 = h2
0kP 3e0 = h20lP
3d0 = h0h2d20P
2i = h0h2d0iP2d0 =
h0h2e0P3j = h0h2iPd2
0 = h0h2jP3e0 = h0h2kP 3d0 =
h0h2Pe0P2j = h0h2PjP 2e0 = h0d
20iP
2h2 = h0d0f0P3d0 =
h0d0Ph2iPd0 = h0d0kP 3h2 = h0e0Ph2P2j = h0e0jP
3h2 =h0e0P
2h2Pj = h0f0Pd0P2d0 = h0gPh2P
2i = h0giP 3h2 =h0Ph2jP
2e0 = h0Ph2kP 2d0 = h0Ph2Pe0Pj = h0jPe0P2h2 =
h0kPd0P2h2 = h0mP 4h2 = h1d
20P
3e0 = h1d0e0P3d0 =
h1d0Pd0P2e0 = h1d0Pe0P
2d0 = h1e0Pd0P2d0 = h1gP 4e0 =
h1Pd20Pe0 = h2
2d0P3j = h2
2iP3e0 = h2
2jP3d0 = h2
2Pd0P2j =
h22Pe0P
2i = h22PjP 2d0 = h2d0Ph2P
2j = h2d0jP3h2 =
h2d0P2h2Pj = h2e0Ph2P
2i = h2e0iP3h2 = h2f0P
4e0 =h2Ph2iP
2e0 = h2Ph2jP2d0 = h2Ph2Pd0Pj = h2iPe0P
2h2 =h2jPd0P
2h2 = h2lP4h2 = c0iP
2i = d20e0P
3h1 = d20Ph1P
2e0 =d20Pe0P
2h1 = d0e0Ph1P2d0 = d0e0Pd0P
2h1 = d0Ph1Pd0Pe0 =d0Ph2
2Pj = d0Ph2jP2h2 = e0f0P
4h2 = e0gP 4h1 = e0Ph1Pd20 =
e0Ph2iP2h2 = f0Ph2P
3e0 = f0Pe0P3h2 = f0P
2h2P2e0 =
gPh1P3e0 = gPe0P
3h1 = gP 2h1P2e0 = Ph2
2iPe0 = Ph22jPd0 =
Ph2lP3h2 = rP 4c0 = i2P 2c0 = lP 2h2
2
32 (1) P 7d0
33 (1) h0P7d0
34 (1) h20P
7d0 = h2P8h2 = Ph2P
7h2 = P 2h2P6h2 = P 3h2P
5h2 = P 4h22
54 ROBERT R. BRUNER
Stem 713 (1) h1h3h6
4 (1) h6c0
5 (1) h1p1 = h2d2
6 (10) x6,47
(01) h21p
′ = h1h25c0
7 (1) h0x6,47 = h3A′′
9 (1) h2G21 = h23Q2
10 (1) h0h2G21 = h0h23Q2 = py
11 (1) h4Q1 = d0Q2
12 (10) x12,37
(01) h0h4Q1 = h0d0Q2 = h22B23 = h2h4x
′ = h2gB2 = h5d0j = h5e0i =Ph2B3
13 (100) g2n = tm
(010) x13,34
(001) x13,35
14 (10) h0x13,34
(01) h0x13,35 = f0x′ = Ph2B4 = iB2
(11) h2P2D1 = e0R1
15 (10) h20x13,34 = h2
1W1 = h1Ph1X1 = h3PQ1 = qu = GP 2h1
(01) h20x13,35 = h0f0x
′ = h0Ph2B4 = h0iB2 = h1e0x′ = Ph1x10,28 =
B1Pe0
(11) h0h2P2D1 = h0e0R1 = h1d0Q1 = h2
2R2 = h2d0R1 = Ph1x10,27
16 (1) d0e0g2 = e3
0g = rz = jw = kv = lu
19 (1) d30k = d2
0e0j = d20gi = d0e
20i = d0lP e0 = d0mPd0 = e0gPj =
e0kPe0 = e0lPd0 = gjPe0 = gkPd0
20 (1) h21x18,20 = h1P
2h1x′ = Ph2
1x′ = B1P
3h1
22 (1) d0iP j = rP 3e0 = i2Pe0 = ijPd0 = kP 2j = lP 2i = zP 2d0
25 (1) P 4u
28 (1) d0P5e0 = e0P
5d0 = Pd0P4e0 = Pe0P
4d0 = P 2d0P3e0 =
P 2e0P3d0
29 (1) h0d0P5e0 = h0e0P
5d0 = h0Pd0P4e0 = h0Pe0P
4d0 =h0P
2d0P3e0 = h0P
2e0P3d0 = h2d0P
5d0 = h2Pd0P4d0 =
h2P2d0P
3d0 = d20P
5h2 = d0Ph2P4d0 = d0Pd0P
4h2 =d0P
2h2P3d0 = d0P
2d0P3h2 = gP 6h2 = Ph2Pd0P
3d0 =Ph2P
2d20 = Pd2
0P3h2 = Pd0P
2h2P2d0
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 55
Stem 71 continued30 (1) h2
0d0P5e0 = h2
0e0P5d0 = h2
0Pd0P4e0 = h2
0Pe0P4d0 =
h20P
2d0P3e0 = h2
0P2e0P
3d0 = h0h2d0P5d0 = h0h2Pd0P
4d0 =h0h2P
2d0P3d0 = h0d
20P
5h2 = h0d0Ph2P4d0 = h0d0Pd0P
4h2 =h0d0P
2h2P3d0 = h0d0P
2d0P3h2 = h0gP 6h2 = h0Ph2Pd0P
3d0 =h0Ph2P
2d20 = h0Pd2
0P3h2 = h0Pd0P
2h2P2d0 = h2
2P6e0 =
h2e0P6h2 = h2Ph2P
5e0 = h2Pe0P5h2 = h2P
2h2P4e0 =
h2P2e0P
4h2 = h2P3h2P
3e0 = e0Ph2P5h2 = e0P
2h2P4h2 =
e0P3h2
2 = Ph22P
4e0 = Ph2Pe0P4h2 = Ph2P
2h2P3e0 =
Ph2P2e0P
3h2 = Pc0P4i = iP 5c0 = Pe0P
2h2P3h2 = P 2h2
2P2e0 =
P 3c0P2i
31 (1) P 6i
32 (1) h0P6i
33 (01) h20P
6i
(11) h1P7d0 = d0P
7h1 = Ph1P6d0 = Pd0P
6h1 = P 2h1P5d0 =
P 2d0P5h1 = P 3h1P
4d0 = P 3d0P4h1
34 (1) h30P
6i
35 (1) h40P
6i
36 (1) h50P
6i
56 ROBERT R. BRUNER
Stem 724 (1) h2
1h3h6 = h32h6
5 (1) h1h6c0
6 (1) h6Ph1
8 (10) h23D2
(11) h4Q2
9 (1) h0h23D2 = h0h4Q2 = h2
2x7,40 = gD1
10 (1) d0D2
11 (1) h0d0D2 = h2PA = Ph2A
12 (1) h20d0D2 = h0h2PA = h0Ph2A = h1h4Q1 = h1d0Q2 = h2
2B5 =h2
2PD2 = h2h4R1 = h2Ph2D2 = g2d1 = Ph1x7,33 = Ph1x7,34 = t2
13 (1) h1x12,37 = c0x10,32
15 (1) e0gm = g2l = rv
18 (10) P 2Q1
(01) d30r = d0e0z = d0im = d0jl = d0k
2 = e0rPe0 = e0il = e0jk =grPd0 = gik = gj2
19 (1) h0P2Q1 = h2x18,20 = P 2h2x
′
20 (1) h20P
2Q1 = h0h2x18,20 = h0P2h2x
′
21 (10) d0P2v = Pd0Pv = vP 2d0
(01) h30P
2Q1 = h20h2x18,20 = h2
0P2h2x
′ = h31x18,20 = h2
1P2h1x
′ =h1Ph2
1x′ = h1B1P
3h1 = Ph1B1P2h1 = Pe0Q
(11) e0P2u = rP 2j = i2j = Pe0Pu = uP 2e0
24 (1) d30P
2d0 = d20Pd2
0 = d0e0P3e0 = d0gP 3d0 = d0Pe0P
2e0 =e20P
3d0 = e0Pd0P2e0 = e0Pe0P
2d0 = gPd0P2d0 = Pd0Pe2
0
26 (1) h1P4u = Ph1P
3u = P 2h1P2u = uP 4h1 = P 3h1Q = P 3h1Pu
27 (1) d0P4j = e0P
4i = iP 4e0 = jP 4d0 = Pd0P3j = PjP 3d0 =
P 2d0P2j = P 2e0P
2i
28 (1) h0d0P4j = h0e0P
4i = h0iP4e0 = h0jP
4d0 = h0Pd0P3j =
h0PjP 3d0 = h0P2d0P
2j = h0P2e0P
2i = h2d0P4i = h2iP
4d0 =h2P
2d0P2i = d0iP
4h2 = d0P2h2P
2i = f0P5d0 = Ph2iP
3d0 =Ph2Pd0P
2i = iPd0P3h2 = iP 2h2P
2d0 = kP 5h2
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 57
Stem 72 continued29 (1) h2
0d0P4j = h2
0e0P4i = h2
0iP4e0 = h2
0jP4d0 = h2
0Pd0P3j =
h20PjP 3d0 = h2
0P2d0P
2j = h20P
2e0P2i = h0h2d0P
4i =h0h2iP
4d0 = h0h2P2d0P
2i = h0d0iP4h2 = h0d0P
2h2P2i =
h0f0P5d0 = h0Ph2iP
3d0 = h0Ph2Pd0P2i = h0iPd0P
3h2 =h0iP
2h2P2d0 = h0kP 5h2 = h1d0P
5e0 = h1e0P5d0 =
h1Pd0P4e0 = h1Pe0P
4d0 = h1P2d0P
3e0 = h1P2e0P
3d0 =h2
2P5j = h2Ph2P
4j = h2jP5h2 = h2P
2h2P3j = h2PjP 4h2 =
h2P3h2P
2j = d0e0P5h1 = d0Ph1P
4e0 = d0Pe0P4h1 =
d0P2h1P
3e0 = d0P2e0P
3h1 = e0Ph1P4d0 = e0Pd0P
4h1 =e0P
2h1P3d0 = e0P
2d0P3h1 = Ph1Pd0P
3e0 = Ph1Pe0P3d0 =
Ph1P2d0P
2e0 = Ph22P
3j = Ph2jP4h2 = Ph2P
2h2P2j =
Ph2PjP 3h2 = jP 2h2P3h2 = Pd0Pe0P
3h1 = Pd0P2h1P
2e0 =Pe0P
2h1P2d0 = P 2h2
2Pj
34 (1) h21P
7d0 = h1d0P7h1 = h1Ph1P
6d0 = h1Pd0P6h1 =
h1P2h1P
5d0 = h1P2d0P
5h1 = h1P3h1P
4d0 = h1P3d0P
4h1 =h3P
8h1 = c0P7c0 = d0Ph1P
6h1 = d0P2h1P
5h1 = d0P3h1P
4h1 =Ph2
1P5d0 = Ph1Pd0P
5h1 = Ph1P2h1P
4d0 = Ph1P2d0P
4h1 =Ph1P
3h1P3d0 = Pc0P
6c0 = Pd0P2h1P
4h1 = Pd0P3h2
1 =P 2h2
1P3d0 = P 2h1P
2d0P3h1 = P 2c0P
5c0 = P 3c0P4c0
35 (1) P 8c0
58 ROBERT R. BRUNER
Stem 737 (100) h4D2
(010) h22Q3 = h3r1 = h2
5Ph2
(001) h1h6Ph1
8 (1) h0h4D2
9 (1) h20h4D2 = h1h4Q2
14 (1) d0B21 = e0Q1 = gx′
15 (1) h0d0B21 = h0e0Q1 = h0gx′ = h2d0Q1 = h2e0x′ = Ph2x10,27 =
Ph2x10,28 = B2Pe0
16 (1) h20d0B21 = h2
0e0Q1 = h20gx′ = h0h2d0Q1 = h0h2e0x
′ =h0Ph2x10,27 = h0Ph2x10,28 = h0B2Pe0 = h2
2d0x′ = h2Ph2B21 =
h2B2Pd0 = c0R2 = d0Ph2B2
17 (10) PR2
(01) d20w = d0e0v = d0gu = d0rk = e2
0u = e0rj = gri = lz
18 (1) h0PR2 = h2R1
19 (1) h20PR2 = h0h2R1 = h1P
2Q1 = Ph1PQ1 = P 2h1Q1 = P 2h2R1
20 (1) d40e0 = d2
0gPe0 = d0e20Pe0 = d0e0gPd0 = e3
0Pd0 = g2P 2e0 =iPv = jPu = uPj
23 (1) d20iPd0 = d0e0P
2j = d0gP 2i = d0jP2e0 = d0kP 2d0 = d0Pe0Pj =
e20P
2i = e0iP2e0 = e0jP
2d0 = e0Pd0Pj = giP 2d0 = iPe20 =
jPd0Pe0 = kPd20 = lP 3e0 = mP 3d0
26 (1) iP 3j = PjP 2i
27 (1) h0iP3j = h0PjP 2i = h2
1P4u = h1Ph1P
3u = h1P2h1P
2u =h1uP 4h1 = h1P
3h1Q = h1P3h1Pu = f0P
4i = Ph21P
2u =Ph1P
2h1Q = Ph1P2h1Pu = Ph1uP 3h1 = Ph2iP
2i = rP 5h2 =qP 5h1 = i2P 3h2 = P 2h2
1u
32 (1) P 7e0
33 (1) h0P7e0 = h2P
7d0 = d0P7h2 = Ph2P
6d0 = Pd0P6h2 =
P 2h2P5d0 = P 2d0P
5h2 = P 3h2P4d0 = P 3d0P
4h2
34 (1) h20P
7e0 = h0h2P7d0 = h0d0P
7h2 = h0Ph2P6d0 = h0Pd0P
6h2 =h0P
2h2P5d0 = h0P
2d0P5h2 = h0P
3h2P4d0 = h0P
3d0P4h2
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 59
Stem 73 continued35 (1) h3
0P7e0 = h2
0h2P7d0 = h2
0d0P7h2 = h2
0Ph2P6d0 = h2
0Pd0P6h2 =
h20P
2h2P5d0 = h2
0P2d0P
5h2 = h20P
3h2P4d0 = h2
0P3d0P
4h2 =h3
1P7d0 = h2
1d0P7h1 = h2
1Ph1P6d0 = h2
1Pd0P6h1 =
h21P
2h1P5d0 = h2
1P2d0P
5h1 = h21P
3h1P4d0 = h2
1P3d0P
4h1 =h1h3P
8h1 = h1c0P7c0 = h1d0Ph1P
6h1 = h1d0P2h1P
5h1 =h1d0P
3h1P4h1 = h1Ph2
1P5d0 = h1Ph1Pd0P
5h1 =h1Ph1P
2h1P4d0 = h1Ph1P
2d0P4h1 = h1Ph1P
3h1P3d0 =
h1Pc0P6c0 = h1Pd0P
2h1P4h1 = h1Pd0P
3h21 = h1P
2h21P
3d0 =h1P
2h1P2d0P
3h1 = h1P2c0P
5c0 = h1P3c0P
4c0 = h22P
8h2 =h2Ph2P
7h2 = h2P2h2P
6h2 = h2P3h2P
5h2 = h2P4h2
2 =h3Ph1P
7h1 = h3P2h1P
6h1 = h3P3h1P
5h1 = h3P4h2
1 =c20P
7h1 = c0Ph1P6c0 = c0Pc0P
6h1 = c0P2h1P
5c0 =c0P
2c0P5h1 = c0P
3h1P4c0 = c0P
3c0P4h1 = d0Ph2
1P5h1 =
d0Ph1P2h1P
4h1 = d0Ph1P3h2
1 = d0P2h2
1P3h1 = Ph3
1P4d0 =
Ph21Pd0P
4h1 = Ph21P
2h1P3d0 = Ph2
1P2d0P
3h1 =Ph1Pc0P
5c0 = Ph1Pd0P2h1P
3h1 = Ph1P2h2
1P2d0 =
Ph1P2c0P
4c0 = Ph1P3c2
0 = Ph22P
6h2 = Ph2P2h2P
5h2 =Ph2P
3h2P4h2 = Pc2
0P5h1 = Pc0P
2h1P4c0 = Pc0P
2c0P4h1 =
Pc0P3h1P
3c0 = Pd0P2h3
1 = P 2h1P2c0P
3c0 = P 2h22P
4h2 =P 2h2P
3h22 = P 2c2
0P3h1
36 (1) h1P8c0 = c0P
8h1 = Ph1P7c0 = Pc0P
7h1 = P 2h1P6c0 =
P 2c0P6h1 = P 3h1P
5c0 = P 3c0P5h1 = P 4h1P
4c0
37 (1) P 9h1
60 ROBERT R. BRUNER
Stem 746 (10) h3n1
(01) h6Ph2
7 (1) h0h6Ph2
8 (10) x8,51
(01) h20h6Ph2 = h2
1h6Ph1
9 (1) h0x8,51
10 (1) h20x8,51 = h3x9,40 = g2r = x2
11 (1) h30x8,51 = h0h3x9,40 = h0g2r = h0x
2 = h23B4 = h4B21 = d0B3 =
e0Q2
12 (1) h40x8,51 = h2
0h3x9,40 = h20g2r = h2
0x2 = h0h
23B4 = h0h4B21 =
h0d0B3 = h0e0Q2 = h2h4Q1 = h2d0Q2 = h3g2i = h3xr =h3x11,35 = h5d0k = h5e0j = h5gi = Ph2x7,33 = Ph2x7,34
13 (1) d0B4
14 (10) gR1
(01) h0d0B4 = h2x13,35 = f0Q1 = jB2
(11) h2x13,34
15 (1) h20d0B4 = h0h2x13,35 = h0f0Q1 = h0gR1 = h0jB2 = h1d0B21 =
h1e0Q1 = h1gx′ = h22P
2D1 = h2e0R1 = h2f0x′ = h2Ph2B4 =
h2iB2 = d20B1 = Ph1B23
16 (1) d0g3 = d0r
2 = e20g
2 = kw = lv = mu
19 (1) d30l = d2
0e0k = d20gj = d0e
20j = d0e0gi = d0mPe0 = e3
0i = e0lP e0 =e0mPd0 = g2Pj = gkPe0 = glPd0
22 (1) d20i
2 = d0rP2d0 = d0jP j = e0iP j = rPd2
0 = ijPe0 = ikPd0 =j2Pd0 = lP 2j = mP 2i = zP 2e0
25 (1) P 4v
28 (1) d20P
4d0 = d0Pd0P3d0 = d0P
2d20 = e0P
5e0 = gP 5d0 = Pd20P
2d0 =Pe0P
4e0 = P 2e0P3e0
29 (1) h0d20P
4d0 = h0d0Pd0P3d0 = h0d0P
2d20 = h0e0P
5e0 = h0gP 5d0 =h0Pd2
0P2d0 = h0Pe0P
4e0 = h0P2e0P
3e0 = h2d0P5e0 =
h2e0P5d0 = h2Pd0P
4e0 = h2Pe0P4d0 = h2P
2d0P3e0 =
h2P2e0P
3d0 = d0e0P5h2 = d0Ph2P
4e0 = d0Pe0P4h2 =
d0P2h2P
3e0 = d0P2e0P
3h2 = e0Ph2P4d0 = e0Pd0P
4h2 =e0P
2h2P3d0 = e0P
2d0P3h2 = Ph2Pd0P
3e0 = Ph2Pe0P3d0 =
Ph2P2d0P
2e0 = Pd0Pe0P3h2 = Pd0P
2h2P2e0 = Pe0P
2h2P2d0
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 61
Stem 74 continued30 (1) h2
0d20P
4d0 = h20d0Pd0P
3d0 = h20d0P
2d20 = h2
0e0P5e0 = h2
0gP 5d0 =h2
0Pd20P
2d0 = h20Pe0P
4e0 = h20P
2e0P3e0 = h0h2d0P
5e0 =h0h2e0P
5d0 = h0h2Pd0P4e0 = h0h2Pe0P
4d0 = h0h2P2d0P
3e0 =h0h2P
2e0P3d0 = h0d0e0P
5h2 = h0d0Ph2P4e0 = h0d0Pe0P
4h2 =h0d0P
2h2P3e0 = h0d0P
2e0P3h2 = h0e0Ph2P
4d0 =h0e0Pd0P
4h2 = h0e0P2h2P
3d0 = h0e0P2d0P
3h2 =h0Ph2Pd0P
3e0 = h0Ph2Pe0P3d0 = h0Ph2P
2d0P2e0 =
h0Pd0Pe0P3h2 = h0Pd0P
2h2P2e0 = h0Pe0P
2h2P2d0 =
h22d0P
5d0 = h22Pd0P
4d0 = h22P
2d0P3d0 = h2d
20P
5h2 =h2d0Ph2P
4d0 = h2d0Pd0P4h2 = h2d0P
2h2P3d0 =
h2d0P2d0P
3h2 = h2gP 6h2 = h2Ph2Pd0P3d0 = h2Ph2P
2d20 =
h2Pd20P
3h2 = h2Pd0P2h2P
2d0 = h4P7h2 = c0P
5j =d20Ph2P
4h2 = d20P
2h2P3h2 = d0Ph2
2P3d0 = d0Ph2Pd0P
3h2 =d0Ph2P
2h2P2d0 = d0Pd0P
2h22 = gPh2P
5h2 = gP 2h2P4h2 =
gP 3h22 = Ph2
2Pd0P2d0 = Ph2Pd2
0P2h2 = Pc0P
4j = jP 5c0 =P 2c0P
3j = PjP 4c0 = P 3c0P2j
31 (1) P 6j
32 (1) h0P6j = h2P
6i = iP 6h2 = P 2h2P4i = P 2iP 4h2
33 (1) h20P
6j = h0h2P6i = h0iP
6h2 = h0P2h2P
4i = h0P2iP 4h2 =
h1P7e0 = e0P
7h1 = Ph1P6e0 = Pe0P
6h1 = P 2h1P5e0 =
P 2e0P5h1 = P 3h1P
4e0 = P 3e0P4h1
38 (1) h1P9h1 = Ph1P
8h1 = P 2h1P7h1 = P 3h1P
6h1 = P 4h1P5h1
62 ROBERT R. BRUNER
Stem 755 (1) h3d2 = h5g2
6 (1) h0h3d2 = h0h5g2 = h23D3
7 (1) x7,53
8 (01) h0x7,53 = h23A
′
(11) h4B3 = c0Q3 = d0D3
9 (01) h20x7,53 = h0h
23A
′ = h1x8,51 = e1x = g2n
(11) h3G21
10 (1) h0h3G21 = d0A = e0D2
11 (1) h20h3G21 = h0d0A = h0e0D2 = h2d0D2 = h2
3X1 = f0Q2 =Ph2A
′′ = xy
15 (1) g2m = rw
18 (10) Pd0x′
(01) d20e0r = d0gz = d0jm = d0kl = e2
0z = e0im = e0jl = e0k2 =
grPe0 = gil = gjk
19 (1) h0Pd0x′ = h2P
2Q1 = Ph2PQ1 = P 2h2Q1
20 (1) h20Pd0x
′ = h0h2P2Q1 = h0Ph2PQ1 = h0P
2h2Q1 = h22x18,20 =
h2P2h2x
′ = Ph22x
′ = B2P3h2
21 (1) d20Pu = d0Pd0u = e0P
2v = gP 2u = riPd0 = i2k = ij2 =Pe0Pv = vP 2e0 = wP 2d0 = zPj
24 (1) d30P
2e0 = d20e0P
2d0 = d20Pd0Pe0 = d0e0Pd2
0 = d0gP 3e0 =e20P
3e0 = e0gP 3d0 = e0Pe0P2e0 = gPd0P
2e0 = gPe0P2d0 = Pe3
0
27 (1) d0iP3d0 = d0Pd0P
2i = e0P4j = gP 4i = iPd0P
2d0 = jP 4e0 =kP 4d0 = Pe0P
3j = PjP 3e0 = P 2e0P2j
28 (1) h0d0iP3d0 = h0d0Pd0P
2i = h0e0P4j = h0gP 4i = h0iPd0P
2d0 =h0jP
4e0 = h0kP 4d0 = h0Pe0P3j = h0PjP 3e0 = h0P
2e0P2j =
h2d0P4j = h2e0P
4i = h2iP4e0 = h2jP
4d0 = h2Pd0P3j =
h2PjP 3d0 = h2P2d0P
2j = h2P2e0P
2i = d0Ph2P3j = d0jP
4h2 =d0P
2h2P2j = d0PjP 3h2 = e0iP
4h2 = e0P2h2P
2i = f0P5e0 =
Ph2iP3e0 = Ph2jP
3d0 = Ph2Pd0P2j = Ph2Pe0P
2i =Ph2PjP 2d0 = iPe0P
3h2 = iP 2h2P2e0 = jPd0P
3h2 =jP 2h2P
2d0 = lP 5h2 = Pd0P2h2Pj
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 63
Stem 75 continued29 (1) h2
0d0iP3d0 = h2
0d0Pd0P2i = h2
0e0P4j = h2
0gP 4i =h2
0iPd0P2d0 = h2
0jP4e0 = h2
0kP 4d0 = h20Pe0P
3j = h20PjP 3e0 =
h20P
2e0P2j = h0h2d0P
4j = h0h2e0P4i = h0h2iP
4e0 =h0h2jP
4d0 = h0h2Pd0P3j = h0h2PjP 3d0 = h0h2P
2d0P2j =
h0h2P2e0P
2i = h0d0Ph2P3j = h0d0jP
4h2 = h0d0P2h2P
2j =h0d0PjP 3h2 = h0e0iP
4h2 = h0e0P2h2P
2i = h0f0P5e0 =
h0Ph2iP3e0 = h0Ph2jP
3d0 = h0Ph2Pd0P2j = h0Ph2Pe0P
2i =h0Ph2PjP 2d0 = h0iPe0P
3h2 = h0iP2h2P
2e0 = h0jPd0P3h2 =
h0jP2h2P
2d0 = h0lP5h2 = h0Pd0P
2h2Pj = h1d20P
4d0 =h1d0Pd0P
3d0 = h1d0P2d2
0 = h1e0P5e0 = h1gP 5d0 =
h1Pd20P
2d0 = h1Pe0P4e0 = h1P
2e0P3e0 = h2
2d0P4i = h2
2iP4d0 =
h22P
2d0P2i = h2d0iP
4h2 = h2d0P2h2P
2i = h2f0P5d0 =
h2Ph2iP3d0 = h2Ph2Pd0P
2i = h2iPd0P3h2 = h2iP
2h2P2d0 =
h2kP 5h2 = d30P
4h1 = d20Ph1P
3d0 = d20Pd0P
3h1 = d20P
2h1P2d0 =
d0f0P5h2 = d0gP 5h1 = d0Ph1Pd0P
2d0 = d0Ph22P
2i =d0Ph2iP
3h2 = d0iP2h2
2 = d0Pd20P
2h1 = e20P
5h1 = e0Ph1P4e0 =
e0Pe0P4h1 = e0P
2h1P3e0 = e0P
2e0P3h1 = f0Ph2P
4d0 =f0Pd0P
4h2 = f0P2h2P
3d0 = f0P2d0P
3h2 = gPh1P4d0 =
gPd0P4h1 = gP 2h1P
3d0 = gP 2d0P3h1 = Ph1Pd3
0 =Ph1Pe0P
3e0 = Ph1P2e2
0 = Ph22iP
2d0 = Ph2iPd0P2h2 =
Ph2kP 4h2 = kP 2h2P3h2 = Pe2
0P3h1 = Pe0P
2h1P2e0
37 (1) P 9h2
38 (1) h0P9h2
39 (1) h20P
9h2 = h21P
9h1 = h1Ph1P8h1 = h1P
2h1P7h1 =
h1P3h1P
6h1 = h1P4h1P
5h1 = Ph21P
7h1 = Ph1P2h1P
6h1 =Ph1P
3h1P5h1 = Ph1P
4h21 = P 2h2
1P5h1 = P 2h1P
3h1P4h1 =
P 3h31
64 ROBERT R. BRUNER
Stem 765 (1) h4D3
6 (1) x6,53
7 (10) h23H1 = h4A
′
(01) h0x6,53
(11) h4A
8 (10) h1x7,53 = d1g2 = e21
(01) h20x6,53 = h0h4A = h2h4D2
9 (10) x9,51
(01) h30x6,53 = h2
0h4A = h0h2h4D2 = h1h4B3 = h1c0Q3 = h1d0D3 =h3PD3 = h2
4B1
10 (1) h0x9,51 = h1h3G21 = h23x8,32 = c1Q2 = f0D2 = e1y = g2q =
Ph1Q3
14 (01) d0x10,27 = d0x10,28 = e0B21 = gQ1
(11) g2t = nw = rN
15 (1) h0d0x10,27 = h0d0x10,28 = h0e0B21 = h0gQ1 = h2d0B21 =h2e0Q1 = h2gx′ = d2
0B2 = Ph2B23
16 (1) x16,32
17 (010) d0e0w = d0gv = d0rl = e20v = e0gu = e0rk = grj = mz
(110) ix′
(001) h0x16,32
18 (10) h0ix′
(01) h20x16,32
(11) h2PR2 = Ph2R2 = Pd0R1
19 (10) h20ix
′
(01) h30x16,32 = h3x18,20
(11) h0h2PR2 = h0Ph2R2 = h0Pd0R1 = h1Pd0x′ = h2
2R1 =d0Ph1x
′ = B1P2d0 = P 2h1B21
20 (10) d40g = d3
0e20 = d0e0gPe0 = d0g
2Pd0 = d0iu = e30Pe0 = e2
0gPd0 =jPv = kPu = vPj
(01) h40x16,32 = h3
0ix′ = h0h3x18,20 = xP 2i
(11) ri2
21 (1) h50x16,32 = h4
0ix′ = h2
0h3x18,20 = h0xP 2i = h0ri2
22 (1) h60x16,32 = h5
0ix′ = h3
0h3x18,20 = h20xP 2i = h2
0ri2 = h3rP
2i = h3i3
23 (1) d30Pj = d2
0iPe0 = d20jPd0 = d0e0iPd0 = d0gP 2j = d0kP 2e0 =
d0lP2d0 = e2
0P2j = e0gP 2i = e0jP
2e0 = e0kP 2d0 = e0Pe0Pj =giP 2e0 = gjP 2d0 = gPd0Pj = jPe2
0 = kPd0Pe0 = lPd20 =
mP 3e0
26 (1) d0iP2i = rP 4d0 = i2P 2d0 = jP 3j = PjP 2j
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 65
Stem 76 continued32 (1) d0P
6d0 = Pd0P5d0 = P 2d0P
4d0 = P 3d20
33 (1) h0d0P6d0 = h0Pd0P
5d0 = h0P2d0P
4d0 = h0P3d2
0 = h2P7e0 =
e0P7h2 = Ph2P
6e0 = Pe0P6h2 = P 2h2P
5e0 = P 2e0P5h2 =
P 3h2P4e0 = P 3e0P
4h2
34 (1) h20d0P
6d0 = h20Pd0P
5d0 = h20P
2d0P4d0 = h2
0P3d2
0 = h0h2P7e0 =
h0e0P7h2 = h0Ph2P
6e0 = h0Pe0P6h2 = h0P
2h2P5e0 =
h0P2e0P
5h2 = h0P3h2P
4e0 = h0P3e0P
4h2 = h22P
7d0 =h2d0P
7h2 = h2Ph2P6d0 = h2Pd0P
6h2 = h2P2h2P
5d0 =h2P
2d0P5h2 = h2P
3h2P4d0 = h2P
3d0P4h2 = d0Ph2P
6h2 =d0P
2h2P5h2 = d0P
3h2P4h2 = Ph2
2P5d0 = Ph2Pd0P
5h2 =Ph2P
2h2P4d0 = Ph2P
2d0P4h2 = Ph2P
3h2P3d0 =
Pd0P2h2P
4h2 = Pd0P3h2
2 = P 2h22P
3d0 = P 2h2P2d0P
3h2
66 ROBERT R. BRUNER
Stem 773 (1) h2
3h6
4 (1) h0h23h6
5 (1) h6d0
6 (10) h1h4D3
(01) h0h6d0
7 (0100) x7,57
(0010) h1x6,53
(1010) m1
(0001) h20h6d0 = h2h6Ph2
8 (100) x8,57
(001) h0x7,57
(011) h0m1 = pg2
9 (01) h20x7,57
(11) h20m1 = h0pg2 = h2
1x7,53 = h1d1g2 = h1e21 = h2x8,51 = h2
3x7,33 =h3d1e1 = h4x8,33 = h5N = f1x
10 (1) h30x7,57
11 (10) gQ2
(01) h40x7,57 = h4x10,27
12 (10) P 2D3
(01) h50x7,57 = h0h4x10,27
13 (10) e0B4
(01) h60x7,57 = h2
0h4x10,27
14 (10) h0e0B4 = h2d0B4 = f0B21 = kB2
(01) h70x7,57 = h3
0h4x10,27
15 (1) h20e0B4 = h0h2d0B4 = h0f0B21 = h0kB2 = h1d0x10,27 =
h1d0x10,28 = h1e0B21 = h1gQ1 = h22x13,35 = h2f0Q1 = h2gR1 =
h2jB2 = d0e0B1 = Ph2B5 = Ph2PD2 = D2P2h2
16 (10) x16,33
(01) e0g3 = e0r
2 = lw = mv
17 (10) h1x16,32
(01) h0x16,33 = iR1
18 (1) h20x16,33 = h0iR1 = h3R1
19 (10) d30m = d2
0e0l = d20gk = d0e
20k = d0e0gj = d0g
2i = e30j = e2
0gi =e0mPe0 = glPe0 = gmPd0 = rPu
(01) h30x16,33 = h2
0iR1 = h0h3R1 = rQ
20 (1) h40x16,33 = h3
0iR1 = h20h3R1 = h0rQ
21 (1) h50x16,33 = h4
0iR1 = h30h3R1 = h2
0rQ = h3iQ = c0x18,20 = yP 2i =x′P 2c0
22 (1) d20ij = d0e0i
2 = d0rP2e0 = d0kPj = d0Pd0z = e0rP
2d0 =e0jP j = giP j = rPd0Pe0 = ikPe0 = ilPd0 = j2Pe0 = jkPd0 =mP 2j
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 67
Stem 77 continued25 (1) d0P
3u = Pd0P2u = uP 3d0 = P 2d0Pu
28 (1) d20P
4e0 = d0e0P4d0 = d0Pd0P
3e0 = d0Pe0P3d0 = d0P
2d0P2e0 =
e0Pd0P3d0 = e0P
2d20 = gP 5e0 = Pd2
0P2e0 = Pd0Pe0P
2d0
31 (1) iP 5d0 = Pd0P4i = P 2iP 3d0
32 (1) h0iP5d0 = h0Pd0P
4i = h0P2iP 3d0 = h2P
6j = Ph2P5j =
jP 6h2 = P 2h2P4j = PjP 5h2 = P 3h2P
3j = P 2jP 4h2
33 (1) h20iP
5d0 = h20Pd0P
4i = h20P
2iP 3d0 = h0h2P6j = h0Ph2P
5j =h0jP
6h2 = h0P2h2P
4j = h0PjP 5h2 = h0P3h2P
3j =h0P
2jP 4h2 = h1d0P6d0 = h1Pd0P
5d0 = h1P2d0P
4d0 =h1P
3d20 = h2
2P6i = h2iP
6h2 = h2P2h2P
4i = h2P2iP 4h2 =
d20P
6h1 = d0Ph1P5d0 = d0Pd0P
5h1 = d0P2h1P
4d0 =d0P
2d0P4h1 = d0P
3h1P3d0 = f0P
7h2 = gP 7h1 =Ph1Pd0P
4d0 = Ph1P2d0P
3d0 = Ph22P
4i = Ph2iP5h2 =
Ph2P3h2P
2i = iP 2h2P4h2 = iP 3h2
2 = Pd20P
4h1 =Pd0P
2h1P3d0 = Pd0P
2d0P3h1 = P 2h1P
2d20 = P 2h2
2P2i
68 ROBERT R. BRUNER
Stem 782 (1) h4h6
3 (1) h0h4h6
4 (1) h20h4h6
5 (1) h30h4h6
6 (010) h1h6d0
(110) t1(001) h4
0h4h6
7 (10) h0t1 = h3x6,47
(01) h50h4h6
8 (100) h4x7,33 = h4x7,34
(001) h60h4h6
(011) h20t1 = h0h3x6,47 = h1m1 = h1x7,57 = h2
3A′′ = c2x = e0D3 = e1f1
(111) h21x6,53
9 (10) x9,55
(01) h70h4h6
10 (01) e0A′ = e0A = gD2
(11) P 2h25
11 (1) h0P2h2
5
12 (1) h20P
2h25
13 (10) h1P2D3 = e0X1 = f0B4 = D3P
2h1 = Ph1PD3 = rB2 = qB1
(01) h30P
2h25
18 (010) d20gr = d0e
20r = d0km = d0l
2 = e0gz = e0jm = e0kl = gim =gjl = gk2
(110) d0PQ1 = Pd0Q1
(001) h21x16,32 = Ph1W1 = P 2h1X1
(011) u2
(111) Pe0x′
19 (1) h0d0PQ1 = h0Pd0Q1 = h0Pe0x′ = h2Pd0x
′ = d0Ph2x′ =
B2P2d0 = P 2h2B21
20 (1) h20d0PQ1 = h2
0Pd0Q1 = h20Pe0x
′ = h0h2Pd0x′ = h0d0Ph2x
′ =h0B2P
2d0 = h0P2h2B21 = h2
2P2Q1 = h2Ph2PQ1 =
h2P2h2Q1 = c0R1 = Ph2
2Q1
21 (1) d20Pv = d0e0Pu = d0rPj = d0iz = d0Pd0v = d0Pe0u = e0Pd0u =
gP 2v = riPe0 = rjPd0 = i2l = ijk = j3 = wP 2e0
23 (1) Ph1x18,20 = x′P 3h1
24 (1) d40Pd0 = d2
0e0P2e0 = d2
0gP 2d0 = d20Pe2
0 = d0e20P
2d0 =d0e0Pd0Pe0 = d0gPd2
0 = e20Pd2
0 = e0gP 3e0 = g2P 3d0 =gPe0P
2e0 = iP 2u = uP 2i
27 (1) d20P
3j = d0iP3e0 = d0jP
3d0 = d0Pd0P2j = d0Pe0P
2i =d0PjP 2d0 = e0iP
3d0 = e0Pd0P2i = gP 4j = iPd0P
2e0 =iPe0P
2d0 = jPd0P2d0 = kP 4e0 = lP 4d0 = Pd2
0Pj
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 69
Stem 78 continued30 (1) iP 4i = P 2i2
31 (1) h0iP4i = h0P
2i2
32 (1) h20iP
4i = h20P
2i2 = h3P6i
33 (1) h30iP
4i = h30P
2i2 = h0h3P6i
34 (1) h40iP
4i = h40P
2i2 = h20h3P
6i
35 (1) h50iP
4i = h50P
2i2 = h30h3P
6i = c0P7d0 = d0P
7c0 = Pc0P6d0 =
Pd0P6c0 = P 2c0P
5d0 = P 2d0P5c0 = P 3c0P
4d0 = P 3d0P4c0
36 (1) P 8d0
37 (1) h0P8d0
38 (1) h20P
8d0 = h2P9h2 = Ph2P
8h2 = P 2h2P7h2 = P 3h2P
6h2 =P 4h2P
5h2
70 ROBERT R. BRUNER
Stem 793 (1) h1h4h6
5 (1) x1
6 (1) h0x1 = h2h4D3 = h25e0
7 (01) h21h6d0 = h3h6Ph1 = h6c
20
(11) h2x6,53 = h4A′′
8 (10) h6Pc0
(01) h0h2x6,53 = h0h4A′′ = h2h4A
9 (1) h20h2x6,53 = h2
0h4A′′ = h0h2h4A = h3
1x6,53 = h1h4x7,33 =h1h4x7,34 = h2
2h4D2 = h33D2 = c0x6,47 = c2y = e0H1 = d2Ph2 =
p1Ph1
10 (1) h1x9,55 = h2x9,51 = f0A′ = f0A
11 (1) h1P2h2
5 = h4x10,32
13 (1) x13,42
14 (1) d0B23 = e0x10,27 = e0x10,28 = gB21
15 (1) h0d0B23 = h0e0x10,27 = h0e0x10,28 = h0gB21 = h2d0x10,27 =h2d0x10,28 = h2e0B21 = h2gQ1 = h4PQ1 = d0e0B2 = Q2Pd0
16 (1) x16,35
17 (010) d0gw = d0rm = e20w = e0gv = e0rl = g2u = grk
(110) d0R2
(001) h0x16,35 = h2x16,32
(111) iQ1 = jx′
18 (10) h0d0R2 = Ph2P2D1 = Pe0R1
(01) h20x16,35 = h0h2x16,32
(11) h0iQ1 = h0jx′ = h2ix
′ = P 2h2B4
19 (10) h20d0R2 = h0Ph2P
2D1 = h0Pe0R1 = h1d0PQ1 = h1Pd0Q1 =h2
2PR2 = h2Ph2R2 = h2Pd0R1 = d0Ph1Q1 = d0Ph2R1 =P 2h1x10,27
(01) h30x16,35 = h2
0h2x16,32 = h31x16,32 = h1Ph1W1 = h1P
2h1X1 =h1u
2 = h3P2Q1 = Ph2
1X1 = qQ = qPu = GP 3h1
(11) h20iQ1 = h2
0jx′ = h0h2ix
′ = h0P2h2B4 = h1Pe0x
′ = e0Ph1x′ =
B1P2e0 = P 2h1x10,28
20 (1) d30e0g = d2
0e30 = d0g
2Pe0 = d0iv = d0ju = e20gPe0 = e0g
2Pd0 =e0iu = rij = kPv = lPu = wPj
23 (1) d40i = d2
0e0Pj = d20jPe0 = d2
0kPd0 = d0e0iPe0 = d0e0jPd0 =d0giPd0 = d0lP
2e0 = d0mP 2d0 = e20iPd0 = e0gP 2j = e0kP 2e0 =
e0lP2d0 = g2P 2i = gjP 2e0 = gkP 2d0 = gPe0Pj = kPe2
0 =lPd0Pe0 = mPd2
0
24 (1) h1Ph1x18,20 = h1x′P 3h1 = Ph1P
2h1x′ = B1P
4h1
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 71
Stem 79 continued26 (1) d0iP
2j = d0jP2i = e0iP
2i = rP 4e0 = i2P 2e0 = ijP 2d0 =iPd0Pj = kP 3j = zP 3d0
29 (01) P 5u
(11) P 4Q
30 (1) h0P4Q
31 (1) h20P
4Q
32 (01) h30P
4Q
(11) d0P6e0 = e0P
6d0 = Pd0P5e0 = Pe0P
5d0 = P 2d0P4e0 =
P 2e0P4d0 = P 3d0P
3e0
33 (01) h40P
4Q
(11) h0d0P6e0 = h0e0P
6d0 = h0Pd0P5e0 = h0Pe0P
5d0 =h0P
2d0P4e0 = h0P
2e0P4d0 = h0P
3d0P3e0 = h2d0P
6d0 =h2Pd0P
5d0 = h2P2d0P
4d0 = h2P3d2
0 = d20P
6h2 = d0Ph2P5d0 =
d0Pd0P5h2 = d0P
2h2P4d0 = d0P
2d0P4h2 = d0P
3h2P3d0 =
gP 7h2 = Ph2Pd0P4d0 = Ph2P
2d0P3d0 = Pd2
0P4h2 =
Pd0P2h2P
3d0 = Pd0P2d0P
3h2 = P 2h2P2d2
0
34 (01) h50P
4Q
(11) h20d0P
6e0 = h20e0P
6d0 = h20Pd0P
5e0 = h20Pe0P
5d0 =h2
0P2d0P
4e0 = h20P
2e0P4d0 = h2
0P3d0P
3e0 = h0h2d0P6d0 =
h0h2Pd0P5d0 = h0h2P
2d0P4d0 = h0h2P
3d20 = h0d
20P
6h2 =h0d0Ph2P
5d0 = h0d0Pd0P5h2 = h0d0P
2h2P4d0 =
h0d0P2d0P
4h2 = h0d0P3h2P
3d0 = h0gP 7h2 = h0Ph2Pd0P4d0 =
h0Ph2P2d0P
3d0 = h0Pd20P
4h2 = h0Pd0P2h2P
3d0 =h0Pd0P
2d0P3h2 = h0P
2h2P2d2
0 = h22P
7e0 = h2e0P7h2 =
h2Ph2P6e0 = h2Pe0P
6h2 = h2P2h2P
5e0 = h2P2e0P
5h2 =h2P
3h2P4e0 = h2P
3e0P4h2 = c0P
6i = e0Ph2P6h2 =
e0P2h2P
5h2 = e0P3h2P
4h2 = Ph22P
5e0 = Ph2Pe0P5h2 =
Ph2P2h2P
4e0 = Ph2P2e0P
4h2 = Ph2P3h2P
3e0 = iP 6c0 =Pe0P
2h2P4h2 = Pe0P
3h22 = P 2h2
2P3e0 = P 2h2P
2e0P3h2 =
P 2c0P4i = P 2iP 4c0
35 (1) h60P
4Q
36 (1) h70P
4Q
37 (10) h1P8d0 = d0P
8h1 = Ph1P7d0 = Pd0P
7h1 = P 2h1P6d0 =
P 2d0P6h1 = P 3h1P
5d0 = P 3d0P5h1 = P 4h1P
4d0
(01) h80P
4Q
38 (1) h90P
4Q
39 (1) h100 P 4Q
72 ROBERT R. BRUNER
Stem 804 (01) h2
1h4h6
(11) e2
5 (10) h6e0
(01) h0e2
6 (01) h20e2 = h1x1
(11) h0h6e0 = h2h6d0
7 (1) h20h6e0 = h0h2h6d0
8 (1) h30h6e0 = h2
0h2h6d0 = h31h6d0 = h1h3h6Ph1 = h1h6c
20 = h2
2h6Ph2
9 (10) h2x8,57 = c1A′ = c1A = f0H1
(01) h1h6Pc0 = h6c0Ph1
10 (1) h6P2h1
12 (1) x12,44
13 (1) gB4
14 (10) x14,42
(01) h0gB4 = h2e0B4 = h4R2 = d0B5 = d0PD2 = f0x10,27 =f0x10,28 = D2Pd0 = iQ2 = lB2
15 (1) h20gB4 = h0h2e0B4 = h0h4R2 = h0d0B5 = h0d0PD2 =
h0f0x10,27 = h0f0x10,28 = h0D2Pd0 = h0iQ2 = h0lB2 =h1d0B23 = h1e0x10,27 = h1e0x10,28 = h1gB21 = h2
2d0B4 =h2f0B21 = h2kB2 = d0f0B2 = d0gB1 = e2
0B1 = Ph2PA = AP 2h2
16 (10) g4 = gr2 = mw
(01) x16,37
17 (1) h0x16,37 = h1x16,35 = h2x16,33 = jR1
19 (1) d20e0m = d2
0gl = d0e20l = d0e0gk = d0g
2j = e30k = e2
0gj = e0g2i =
gmPe0 = rPv = uz
22 (10) P 3Q1
(01) d20rPd0 = d2
0ik = d20j
2 = d0e0ij = d0gi2 = d0lP j = d0Pe0z =e20i
2 = e0rP2e0 = e0kPj = e0Pd0z = grP 2d0 = gjPj = rPe2
0 =ilPe0 = imPd0 = jkPe0 = jlPd0 = k2Pd0
23 (1) h0P3Q1 = Ph2x18,20 = x′P 3h2
24 (1) h20P
3Q1 = h0Ph2x18,20 = h0x′P 3h2
25 (10) d0P3v = Pd0P
2v = vP 3d0 = P 2d0Pv
(01) h30P
3Q1 = h20Ph2x18,20 = h2
0x′P 3h2 = h2
1Ph1x18,20 =h2
1x′P 3h1 = h1Ph1P
2h1x′ = h1B1P
4h1 = Ph31x
′ = Ph1B1P3h1 =
B1P2h2
1 = P 2e0Q
(11) e0P3u = rP 3j = i2Pj = Pe0P
2u = uP 3e0 = zP 2i = P 2e0Pu
28 (1) d30P
3d0 = d20Pd0P
2d0 = d0e0P4e0 = d0gP 4d0 = d0Pd3
0 =d0Pe0P
3e0 = d0P2e2
0 = e20P
4d0 = e0Pd0P3e0 = e0Pe0P
3d0 =e0P
2d0P2e0 = gPd0P
3d0 = gP 2d20 = Pd0Pe0P
2e0 = Pe20P
2d0
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 73
Stem 80 continued30 (1) h1P
4Q = h1P5u = Ph1P
4u = P 2h1P3u = uP 5h1 = P 3h1P
2u =QP 4h1 = PuP 4h1
31 (1) d0P5j = iP 5e0 = jP 5d0 = Pd0P
4j = Pe0P4i = PjP 4d0 =
P 2d0P3j = P 2iP 3e0 = P 2jP 3d0
32 (1) h0d0P5j = h0iP
5e0 = h0jP5d0 = h0Pd0P
4j = h0Pe0P4i =
h0PjP 4d0 = h0P2d0P
3j = h0P2iP 3e0 = h0P
2jP 3d0 =h2iP
5d0 = h2Pd0P4i = h2P
2iP 3d0 = d0Ph2P4i = d0iP
5h2 =d0P
3h2P2i = f0P
6d0 = Ph2iP4d0 = Ph2P
2d0P2i = iPd0P
4h2 =iP 2h2P
3d0 = iP 2d0P3h2 = kP 6h2 = Pd0P
2h2P2i
33 (1) h20d0P
5j = h20iP
5e0 = h20jP
5d0 = h20Pd0P
4j = h20Pe0P
4i =h2
0PjP 4d0 = h20P
2d0P3j = h2
0P2iP 3e0 = h2
0P2jP 3d0 =
h0h2iP5d0 = h0h2Pd0P
4i = h0h2P2iP 3d0 = h0d0Ph2P
4i =h0d0iP
5h2 = h0d0P3h2P
2i = h0f0P6d0 = h0Ph2iP
4d0 =h0Ph2P
2d0P2i = h0iPd0P
4h2 = h0iP2h2P
3d0 = h0iP2d0P
3h2 =h0kP 6h2 = h0Pd0P
2h2P2i = h1d0P
6e0 = h1e0P6d0 =
h1Pd0P5e0 = h1Pe0P
5d0 = h1P2d0P
4e0 = h1P2e0P
4d0 =h1P
3d0P3e0 = h2
2P6j = h2Ph2P
5j = h2jP6h2 = h2P
2h2P4j =
h2PjP 5h2 = h2P3h2P
3j = h2P2jP 4h2 = d0e0P
6h1 =d0Ph1P
5e0 = d0Pe0P5h1 = d0P
2h1P4e0 = d0P
2e0P4h1 =
d0P3h1P
3e0 = e0Ph1P5d0 = e0Pd0P
5h1 = e0P2h1P
4d0 =e0P
2d0P4h1 = e0P
3h1P3d0 = Ph1Pd0P
4e0 = Ph1Pe0P4d0 =
Ph1P2d0P
3e0 = Ph1P2e0P
3d0 = Ph22P
4j = Ph2jP5h2 =
Ph2P2h2P
3j = Ph2PjP 4h2 = Ph2P3h2P
2j = jP 2h2P4h2 =
jP 3h22 = Pd0Pe0P
4h1 = Pd0P2h1P
3e0 = Pd0P2e0P
3h1 =Pe0P
2h1P3d0 = Pe0P
2d0P3h1 = P 2h1P
2d0P2e0 = P 2h2
2P2j =
P 2h2PjP 3h2
38 (1) h21P
8d0 = h1d0P8h1 = h1Ph1P
7d0 = h1Pd0P7h1 =
h1P2h1P
6d0 = h1P2d0P
6h1 = h1P3h1P
5d0 = h1P3d0P
5h1 =h1P
4h1P4d0 = h3P
9h1 = c0P8c0 = d0Ph1P
7h1 = d0P2h1P
6h1 =d0P
3h1P5h1 = d0P
4h21 = Ph2
1P6d0 = Ph1Pd0P
6h1 =Ph1P
2h1P5d0 = Ph1P
2d0P5h1 = Ph1P
3h1P4d0 =
Ph1P3d0P
4h1 = Pc0P7c0 = Pd0P
2h1P5h1 = Pd0P
3h1P4h1 =
P 2h21P
4d0 = P 2h1P2d0P
4h1 = P 2h1P3h1P
3d0 = P 2c0P6c0 =
P 2d0P3h2
1 = P 3c0P5c0 = P 4c2
0
39 (1) P 9c0
74 ROBERT R. BRUNER
Stem 813 (1) h2h4h6
4 (1) h0h2h4h6
5 (10) h6f0
(01) h20h2h4h6 = h3
1h4h6
6 (1) h0h6f0 = h1h6e0
7 (1) h2t1 = h23n1 = h4r1 = c2f1
8 (1) h4x7,40 = c1H1 = gD3
9 (1) h3x8,51 = d0Q3 = g2x
10 (10) gA′
(01) h0h3x8,51 = h0d0Q3 = h0g2x = h2x9,55 = e0A′′
(11) gA
11 (10) h1h6P2h1 = h6Ph2
1
(01) h0gA′ = h0gA = h2e0A′ = h2e0A = h2gD2 = h4B5 = h4PD2 =
c1x8,32 = c1x8,33 = f0x7,34 = nC
12 (1) x12,45
13 (1) h0x12,45 = h1x12,44
15 (10) gnr = tw = mN
(01) h1x14,42
18 (01) d0e0gr = d0lm = e30r = e0km = e0l
2 = g2z = gjm = gkl = uv
(11) d20x
′ = e0PQ1 = Pd0B21 = Pe0Q1
19 (1) h0d20x
′ = h0e0PQ1 = h0Pd0B21 = h0Pe0Q1 = h2d0PQ1 =h2Pd0Q1 = h2Pe0x
′ = d0Ph2Q1 = e0Ph2x′ = B2P
2e0 =P 2h2x10,27 = P 2h2x10,28
20 (1) h20d
20x
′ = h20e0PQ1 = h2
0Pd0B21 = h20Pe0Q1 = h0h2d0PQ1 =
h0h2Pd0Q1 = h0h2Pe0x′ = h0d0Ph2Q1 = h0e0Ph2x
′ =h0B2P
2e0 = h0P2h2x10,27 = h0P
2h2x10,28 = h22Pd0x
′ =h2d0Ph2x
′ = h2B2P2d0 = h2P
2h2B21 = c0PR2 = d0B2P2h2 =
Ph22B21 = Ph2B2Pd0 = Pc0R2
21 (10) P 2R2
(01) d30u = d2
0ri = d0e0Pv = d0gPu = d0jz = d0Pd0w = d0Pe0v =e20Pu = e0rPj = e0iz = e0Pd0v = e0Pe0u = gPd0u = rjPe0 =
rkPd0 = i2m = ijl = ik2 = j2k
22 (1) h0P2R2 = Ph2R1
23 (1) h20P
2R2 = h0Ph2R1 = h1P3Q1 = Ph1P
2Q1 = P 2h1PQ1 =R1P
3h2 = Q1 P 3h1
24 (1) d40Pe0 = d3
0e0Pd0 = d20gP 2e0 = d0e
20P
2e0 = d0e0gP 2d0 =d0e0Pe2
0 = d0gPd0Pe0 = e30P
2d0 = e20Pd0Pe0 = e0gPd2
0 =g2P 3e0 = iP 2v = jP 2u = uP 2j = vP 2i = PjPu
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 75
Stem 81 continued27 (1) d3
0P2i = d2
0iP2d0 = d0e0P
3j = d0iPd20 = d0jP
3e0 = d0kP 3d0 =d0Pe0P
2j = d0PjP 2e0 = e0iP3e0 = e0jP
3d0 = e0Pd0P2j =
e0Pe0P2i = e0PjP 2d0 = giP 3d0 = gPd0P
2i = iPe0P2e0 =
jPd0P2e0 = jPe0P
2d0 = kPd0P2d0 = lP 4e0 = mP 4d0 =
Pd0Pe0Pj
30 (1) iP 4j = jP 4i = P 2iP 2j
31 (1) h0iP4j = h0jP
4i = h0P2iP 2j = h2
1P4Q = h2
1P5u =
h1Ph1P4u = h1P
2h1P3u = h1uP 5h1 = h1P
3h1P2u =
h1QP 4h1 = h1PuP 4h1 = h2iP4i = h2P
2i2 = Ph21P
3u =Ph1P
2h1P2u = Ph1uP 4h1 = Ph1P
3h1Q = Ph1P3h1Pu =
rP 6h2 = qP 6h1 = i2P 4h2 = iP 2h2P2i = P 2h2
1Q = P 2h21Pu =
P 2h1uP 3h1
36 (1) P 8e0
37 (1) h0P8e0 = h2P
8d0 = d0P8h2 = Ph2P
7d0 = Pd0P7h2 =
P 2h2P6d0 = P 2d0P
6h2 = P 3h2P5d0 = P 3d0P
5h2 = P 4h2P4d0
38 (1) h20P
8e0 = h0h2P8d0 = h0d0P
8h2 = h0Ph2P7d0 = h0Pd0P
7h2 =h0P
2h2P6d0 = h0P
2d0P6h2 = h0P
3h2P5d0 = h0P
3d0P5h2 =
h0P4h2P
4d0
continued
76 ROBERT R. BRUNER
Stem 81 continued39 (1) h3
0P8e0 = h2
0h2P8d0 = h2
0d0P8h2 = h2
0Ph2P7d0 = h2
0Pd0P7h2 =
h20P
2h2P6d0 = h2
0P2d0P
6h2 = h20P
3h2P5d0 = h2
0P3d0P
5h2 =h2
0P4h2P
4d0 = h31P
8d0 = h21d0P
8h1 = h21Ph1P
7d0 =h2
1Pd0P7h1 = h2
1P2h1P
6d0 = h21P
2d0P6h1 = h2
1P3h1P
5d0 =h2
1P3d0P
5h1 = h21P
4h1P4d0 = h1h3P
9h1 = h1c0P8c0 =
h1d0Ph1P7h1 = h1d0P
2h1P6h1 = h1d0P
3h1P5h1 = h1d0P
4h21 =
h1Ph21P
6d0 = h1Ph1Pd0P6h1 = h1Ph1P
2h1P5d0 =
h1Ph1P2d0P
5h1 = h1Ph1P3h1P
4d0 = h1Ph1P3d0P
4h1 =h1Pc0P
7c0 = h1Pd0P2h1P
5h1 = h1Pd0P3h1P
4h1 =h1P
2h21P
4d0 = h1P2h1P
2d0P4h1 = h1P
2h1P3h1P
3d0 =h1P
2c0P6c0 = h1P
2d0P3h2
1 = h1P3c0P
5c0 = h1P4c2
0 = h22P
9h2 =h2Ph2P
8h2 = h2P2h2P
7h2 = h2P3h2P
6h2 = h2P4h2P
5h2 =h3Ph1P
8h1 = h3P2h1P
7h1 = h3P3h1P
6h1 = h3P4h1P
5h1 =c20P
8h1 = c0Ph1P7c0 = c0Pc0P
7h1 = c0P2h1P
6c0 =c0P
2c0P6h1 = c0P
3h1P5c0 = c0P
3c0P5h1 = c0P
4h1P4c0 =
d0Ph21P
6h1 = d0Ph1P2h1P
5h1 = d0Ph1P3h1P
4h1 =d0P
2h21P
4h1 = d0P2h1P
3h21 = Ph3
1P5d0 = Ph2
1Pd0P5h1 =
Ph21P
2h1P4d0 = Ph2
1P2d0P
4h1 = Ph21P
3h1P3d0 =
Ph1Pc0P6c0 = Ph1Pd0P
2h1P4h1 = Ph1Pd0P
3h21 =
Ph1P2h2
1P3d0 = Ph1P
2h1P2d0P
3h1 = Ph1P2c0P
5c0 =Ph1P
3c0P4c0 = Ph2
2P7h2 = Ph2P
2h2P6h2 = Ph2P
3h2P5h2 =
Ph2P4h2
2 = Pc20P
6h1 = Pc0P2h1P
5c0 = Pc0P2c0P
5h1 =Pc0P
3h1P4c0 = Pc0P
3c0P4h1 = Pd0P
2h21P
3h1 = P 2h31P
2d0 =P 2h1P
2c0P4c0 = P 2h1P
3c20 = P 2h2
2P5h2 = P 2h2P
3h2P4h2 =
P 2c20P
4h1 = P 2c0P3h1P
3c0 = P 3h32
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 77
Stem 824 (1) h6c1
6 (10) h4Q3
(01) h2x1 = h23d2 = h3h5g2 = h4n1 = h2
5g = c22
7 (10) h0h4Q3
(01) h0h2x1 = h0h23d2 = h0h3h5g2 = h0h4n1 = h0h
25g = h0c
22 =
h22h4D3 = h2h
25e0 = h3
3D3 = h24D1
8 (10) e1g2
(01) h20h4Q3 = h2
2x6,53 = h2h4A′′ = d0d2
(11) h3x7,53
9 (1) gH1
10 (1) h6P2h2
11 (1) h0h6P2h2
12 (10) gx8,33
(01) h20h6P
2h2 = h21h6P
2h1 = h1h6Ph21
14 (01) e0B23 = gx10,27 = gx10,28
(11) h2x13,42
16 (1) x16,38
17 (010) iB21 = jQ1 = kx′ = Pd0B4
(001) h0x16,38 = h2x16,35
(011) d0P2D1 = e0R2
(111) e0gw = e0rm = g2v = grl
18 (10) h0d0P2D1 = h0e0R2 = h2d0R2 = d2
0R1
(01) h20x16,38 = h0h2x16,35 = h2
2x16,32 = Ph2x13,34
(11) h0iB21 = h0jQ1 = h0kx′ = h0Pd0B4 = h2iQ1 = h2jx′ =
f0PQ1 = Ph2x13,35 = B2Pj
19 (1) h20d0P
2D1 = h20e0R2 = h2
0iB21 = h20jQ1 = h2
0kx′ = h20Pd0B4 =
h0h2d0R2 = h0h2iQ1 = h0h2jx′ = h0d
20R1 = h0f0PQ1 =
h0Ph2x13,35 = h0B2Pj = h1d20x
′ = h1e0PQ1 = h1Pd0B21 =h1Pe0Q1 = h2
2ix′ = h2Ph2P
2D1 = h2Pe0R1 = h2P2h2B4 =
d0Ph1B21 = d0B1Pd0 = e0Ph1Q1 = e0Ph2R1 = f0Ph2x′ =
gPh1x′ = Ph2
2B4 = Ph2iB2 = P 2h1B23
20 (1) d30g
2 = d20e
20g = d0e
40 = d0iw = d0jv = d0ku = e0g
2Pe0 = e0iv =e0ju = g3Pd0 = giu = r2Pd0 = rik = rj2 = lPv = mPu = z2
23 (1) d40j = d3
0e0i = d20gPj = d2
0kPe0 = d20lPd0 = d0e
20Pj = d0e0jPe0 =
d0e0kPd0 = d0giPe0 = d0gjPd0 = d0mP 2e0 = e20iPe0 =
e20jPd0 = e0giPd0 = e0lP
2e0 = e0mP 2d0 = g2P 2j = gkP 2e0 =glP 2d0 = lP e2
0 = mPd0Pe0
26 (1) d0rP3d0 = d0i
2Pd0 = d0jP2j = d0kP 2i = d0Pj2 = e0iP
2j =e0jP
2i = giP 2i = rPd0P2d0 = ijP 2e0 = ikP 2d0 = iPe0Pj =
j2P 2d0 = jPd0Pj = lP 3j = zP 3e0
continued
78 ROBERT R. BRUNER
Stem 82 continued29 (1) P 5v
32 (1) d20P
5d0 = d0Pd0P4d0 = d0P
2d0P3d0 = e0P
6e0 = gP 6d0 =Pd2
0P3d0 = Pd0P
2d20 = Pe0P
5e0 = P 2e0P4e0 = P 3e2
0
33 (1) h0d20P
5d0 = h0d0Pd0P4d0 = h0d0P
2d0P3d0 = h0e0P
6e0 =h0gP 6d0 = h0Pd2
0P3d0 = h0Pd0P
2d20 = h0Pe0P
5e0 =h0P
2e0P4e0 = h0P
3e20 = h2d0P
6e0 = h2e0P6d0 = h2Pd0P
5e0 =h2Pe0P
5d0 = h2P2d0P
4e0 = h2P2e0P
4d0 = h2P3d0P
3e0 =d0e0P
6h2 = d0Ph2P5e0 = d0Pe0P
5h2 = d0P2h2P
4e0 =d0P
2e0P4h2 = d0P
3h2P3e0 = e0Ph2P
5d0 = e0Pd0P5h2 =
e0P2h2P
4d0 = e0P2d0P
4h2 = e0P3h2P
3d0 = Ph2Pd0P4e0 =
Ph2Pe0P4d0 = Ph2P
2d0P3e0 = Ph2P
2e0P3d0 = Pd0Pe0P
4h2 =Pd0P
2h2P3e0 = Pd0P
2e0P3h2 = Pe0P
2h2P3d0 =
Pe0P2d0P
3h2 = P 2h2P2d0P
2e0
34 (1) h20d
20P
5d0 = h20d0Pd0P
4d0 = h20d0P
2d0P3d0 = h2
0e0P6e0 =
h20gP 6d0 = h2
0Pd20P
3d0 = h20Pd0P
2d20 = h2
0Pe0P5e0 =
h20P
2e0P4e0 = h2
0P3e2
0 = h0h2d0P6e0 = h0h2e0P
6d0 =h0h2Pd0P
5e0 = h0h2Pe0P5d0 = h0h2P
2d0P4e0 =
h0h2P2e0P
4d0 = h0h2P3d0P
3e0 = h0d0e0P6h2 =
h0d0Ph2P5e0 = h0d0Pe0P
5h2 = h0d0P2h2P
4e0 =h0d0P
2e0P4h2 = h0d0P
3h2P3e0 = h0e0Ph2P
5d0 =h0e0Pd0P
5h2 = h0e0P2h2P
4d0 = h0e0P2d0P
4h2 =h0e0P
3h2P3d0 = h0Ph2Pd0P
4e0 = h0Ph2Pe0P4d0 =
h0Ph2P2d0P
3e0 = h0Ph2P2e0P
3d0 = h0Pd0Pe0P4h2 =
h0Pd0P2h2P
3e0 = h0Pd0P2e0P
3h2 = h0Pe0P2h2P
3d0 =h0Pe0P
2d0P3h2 = h0P
2h2P2d0P
2e0 = h22d0P
6d0 =h2
2Pd0P5d0 = h2
2P2d0P
4d0 = h22P
3d20 = h2d
20P
6h2 =h2d0Ph2P
5d0 = h2d0Pd0P5h2 = h2d0P
2h2P4d0 =
h2d0P2d0P
4h2 = h2d0P3h2P
3d0 = h2gP 7h2 = h2Ph2Pd0P4d0 =
h2Ph2P2d0P
3d0 = h2Pd20P
4h2 = h2Pd0P2h2P
3d0 =h2Pd0P
2d0P3h2 = h2P
2h2P2d2
0 = h4P8h2 = c0P
6j =d20Ph2P
5h2 = d20P
2h2P4h2 = d2
0P3h2
2 = d0Ph22P
4d0 =d0Ph2Pd0P
4h2 = d0Ph2P2h2P
3d0 = d0Ph2P2d0P
3h2 =d0Pd0P
2h2P3h2 = d0P
2h22P
2d0 = gPh2P6h2 = gP 2h2P
5h2 =gP 3h2P
4h2 = Ph22Pd0P
3d0 = Ph22P
2d20 = Ph2Pd2
0P3h2 =
Ph2Pd0P2h2P
2d0 = Pc0P5j = jP 6c0 = Pd2
0P2h2
2 = P 2c0P4j =
PjP 5c0 = P 3c0P3j = P 2jP 4c0
35 (1) P 7j
36 (1) h0P7j = Ph2P
6i = iP 7h2 = P 3h2P4i = P 2iP 5h2
37 (1) h20P
7j = h0Ph2P6i = h0iP
7h2 = h0P3h2P
4i = h0P2iP 5h2 =
h1P8e0 = e0P
8h1 = Ph1P7e0 = Pe0P
7h1 = P 2h1P6e0 =
P 2e0P6h1 = P 3h1P
5e0 = P 3e0P5h1 = P 4h1P
4e0
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 79
Stem 835 (10) h6g
(01) h2e2 = h4d2
6 (1) h0h6g = h2h6e0
7 (10) h1h4Q3 = h3x6,53
(01) h20h6g = h0h2h6e0 = h2
2h6d0
10 (1) h1gH1 = h22x8,57 = h2c1A
′ = h2c1A = h2f0H1 = h3x9,51 = nD1
11 (1) e0x7,40 = gx7,34
13 (1) h2x12,44
15 (1) x15,41
16 (10) rx′ = iB4
(01) h0x15,41
17 (10) h0rx′ = h0iB4 = h3x16,32
(01) h20x15,41
(11) h1x16,38 = h2x16,37 = d0W1 = f0R2 = kR1
18 (1) h30x15,41 = h2
0rx′ = h2
0iB4 = h0h3x16,32 = h3ix′
19 (10) d20gm = d0e
20m = d0e0gl = d0g
2k = d0ru = e30l = e2
0gk = e0g2j =
g3i = vz
(01) h40x15,41 = h3
0rx′ = h3
0iB4 = h20h3x16,32 = h0h3ix
′ = g2P2i = xi2
(11) r2i
20 (1) h50x15,41 = h4
0rx′ = h4
0iB4 = h30h3x16,32 = h2
0h3ix′ = h0g2P
2i =h0xi2 = h0r
2i = h23x18,20
22 (10) x′P 2d0
(01) d30z = d2
0rPe0 = d20il = d2
0jk = d0e0rPd0 = d0e0ik = d0e0j2 =
d0gij = d0mPj = e20ij = e0gi2 = e0lP j = e0Pe0z = grP 2e0 =
gkPj = gPd0z = imPe0 = jlPe0 = jmPd0 = k2Pe0 = klPd0
(11) d0x18,20
23 (1) h0d0x18,20 = h0x′P 2d0 = h2P
3Q1 = Ph2P2Q1 = P 2h2PQ1 =
Q1 P 3h2
24 (1) h20d0x18,20 = h2
0x′P 2d0 = h0h2P
3Q1 = h0Ph2P2Q1 =
h0P2h2PQ1 = h0Q1 P 3h2 = h2Ph2x18,20 = h2x
′P 3h2 =Ph2P
2h2x′ = B2P
4h2
25 (1) d20P
2u = d0rP2i = d0i
3 = d0Pd0Pu = d0uP 2d0 = e0P3v =
gP 3u = riP 2d0 = ijP j = Pd20u = Pe0P
2v = vP 3e0 = wP 3d0 =zP 2j = P 2e0Pv
28 (1) d30P
3e0 = d20e0P
3d0 = d20Pd0P
2e0 = d20Pe0P
2d0 =d0e0Pd0P
2d0 = d0gP 4e0 = d0Pd20Pe0 = e2
0P4e0 = e0gP 4d0 =
e0Pd30 = e0Pe0P
3e0 = e0P2e2
0 = gPd0P3e0 = gPe0P
3d0 =gP 2d0P
2e0 = Pe20P
2e0
continued
80 ROBERT R. BRUNER
Stem 83 continued31 (1) d2
0P4i = d0iP
4d0 = d0P2d0P
2i = e0P5j = iPd0P
3d0 = iP 2d20 =
jP 5e0 = kP 5d0 = Pd20P
2i = Pe0P4j = PjP 4e0 = P 2e0P
3j =P 2jP 3e0
32 (1) h0d20P
4i = h0d0iP4d0 = h0d0P
2d0P2i = h0e0P
5j =h0iPd0P
3d0 = h0iP2d2
0 = h0jP5e0 = h0kP 5d0 = h0Pd2
0P2i =
h0Pe0P4j = h0PjP 4e0 = h0P
2e0P3j = h0P
2jP 3e0 = h2d0P5j =
h2iP5e0 = h2jP
5d0 = h2Pd0P4j = h2Pe0P
4i = h2PjP 4d0 =h2P
2d0P3j = h2P
2iP 3e0 = h2P2jP 3d0 = d0Ph2P
4j =d0jP
5h2 = d0P2h2P
3j = d0PjP 4h2 = d0P3h2P
2j = e0Ph2P4i =
e0iP5h2 = e0P
3h2P2i = f0P
6e0 = Ph2iP4e0 = Ph2jP
4d0 =Ph2Pd0P
3j = Ph2PjP 3d0 = Ph2P2d0P
2j = Ph2P2e0P
2i =iPe0P
4h2 = iP 2h2P3e0 = iP 2e0P
3h2 = jPd0P4h2 =
jP 2h2P3d0 = jP 2d0P
3h2 = lP 6h2 = Pd0P2h2P
2j =Pd0PjP 3h2 = Pe0P
2h2P2i = P 2h2PjP 2d0
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 81
Stem 83 continued33 (1) h2
0d20P
4i = h20d0iP
4d0 = h20d0P
2d0P2i = h2
0e0P5j =
h20iPd0P
3d0 = h20iP
2d20 = h2
0jP5e0 = h2
0kP 5d0 = h20Pd2
0P2i =
h20Pe0P
4j = h20PjP 4e0 = h2
0P2e0P
3j = h20P
2jP 3e0 =h0h2d0P
5j = h0h2iP5e0 = h0h2jP
5d0 = h0h2Pd0P4j =
h0h2Pe0P4i = h0h2PjP 4d0 = h0h2P
2d0P3j = h0h2P
2iP 3e0 =h0h2P
2jP 3d0 = h0d0Ph2P4j = h0d0jP
5h2 = h0d0P2h2P
3j =h0d0PjP 4h2 = h0d0P
3h2P2j = h0e0Ph2P
4i = h0e0iP5h2 =
h0e0P3h2P
2i = h0f0P6e0 = h0Ph2iP
4e0 = h0Ph2jP4d0 =
h0Ph2Pd0P3j = h0Ph2PjP 3d0 = h0Ph2P
2d0P2j =
h0Ph2P2e0P
2i = h0iPe0P4h2 = h0iP
2h2P3e0 = h0iP
2e0P3h2 =
h0jPd0P4h2 = h0jP
2h2P3d0 = h0jP
2d0P3h2 = h0lP
6h2 =h0Pd0P
2h2P2j = h0Pd0PjP 3h2 = h0Pe0P
2h2P2i =
h0P2h2PjP 2d0 = h1d
20P
5d0 = h1d0Pd0P4d0 = h1d0P
2d0P3d0 =
h1e0P6e0 = h1gP 6d0 = h1Pd2
0P3d0 = h1Pd0P
2d20 =
h1Pe0P5e0 = h1P
2e0P4e0 = h1P
3e20 = h2
2iP5d0 =
h22Pd0P
4i = h22P
2iP 3d0 = h2d0Ph2P4i = h2d0iP
5h2 =h2d0P
3h2P2i = h2f0P
6d0 = h2Ph2iP4d0 = h2Ph2P
2d0P2i =
h2iPd0P4h2 = h2iP
2h2P3d0 = h2iP
2d0P3h2 = h2kP 6h2 =
h2Pd0P2h2P
2i = d30P
5h1 = d20Ph1P
4d0 = d20Pd0P
4h1 =d20P
2h1P3d0 = d2
0P2d0P
3h1 = d0f0P6h2 = d0gP 6h1 =
d0Ph1Pd0P3d0 = d0Ph1P
2d20 = d0Ph2iP
4h2 = d0Ph2P2h2P
2i =d0iP
2h2P3h2 = d0Pd2
0P3h1 = d0Pd0P
2h1P2d0 = e2
0P6h1 =
e0Ph1P5e0 = e0Pe0P
5h1 = e0P2h1P
4e0 = e0P2e0P
4h1 =e0P
3h1P3e0 = f0Ph2P
5d0 = f0Pd0P5h2 = f0P
2h2P4d0 =
f0P2d0P
4h2 = f0P3h2P
3d0 = gPh1P5d0 = gPd0P
5h1 =gP 2h1P
4d0 = gP 2d0P4h1 = gP 3h1P
3d0 = Ph1Pd20P
2d0 =Ph1Pe0P
4e0 = Ph1P2e0P
3e0 = Ph22iP
3d0 = Ph22Pd0P
2i =Ph2iPd0P
3h2 = Ph2iP2h2P
2d0 = Ph2kP 5h2 = iPd0P2h2
2 =kP 2h2P
4h2 = kP 3h22 = Pd3
0P2h1 = Pe2
0P4h1 = Pe0P
2h1P3e0 =
Pe0P2e0P
3h1 = P 2h1P2e2
0
82 ROBERT R. BRUNER
Stem 844 (10) f2
(01) h22h4h6 = h3
3h6
5 (1) h0f2 = h4p′
6 (01) h20f2 = h0h4p
′
(11) h1h6g = h2h6f0
7 (1) h30f2 = h2
0h4p′
8 (1) h40f2 = h3
0h4p′ = h2
1h4Q3 = h1h3x6,53 = h3m1 = h3x7,57 = h24G =
h4c0D3 = d0p1 = f1g2
9 (1) h2h4x7,40 = h2c1H1 = h2gD3 = h3x8,57 = h4PD3 = c0x6,53 =e0Q3 = d1D1
10 (1) Px6,53
14 (1) x14,46
15 (10) x15,42
(01) x15,43
16 (1) h0x15,42 = rR1 = iX1
17 (1) h20x15,42 = h0rR1 = h0iX1 = h3x16,33
18 (001) h30x15,42 = h2
0rR1 = h20iX1 = h0h3x16,33 = h3iR1 = xQ
(011) d20Q1 = d0e0x
′ = gPQ1 = Pd0x10,27 = Pd0x10,28 = Pe0B21
(111) d0g2r = d0m
2 = e20gr = e0lm = gkm = gl2 = uw = v2
19 (10) h4x18,20 = c0x16,32
(01) h40x15,42 = h3
0rR1 = h30iX1 = h2
0h3x16,33 = h0h3iR1 = h0xQ =h2
3R1
(11) h0d20Q1 = h0d0e0x
′ = h0gPQ1 = h0Pd0x10,27 = h0Pd0x10,28 =h0Pe0B21 = h2d
20x
′ = h2e0PQ1 = h2Pd0B21 = h2Pe0Q1 =d0Ph2B21 = d0B2Pd0 = e0Ph2Q1 = gPh2x
′ = P 2h2B23
20 (1) h50x15,42 = h4
0rR1 = h40iX1 = h3
0h3x16,33 = h20h3iR1 = h2
0d20Q1 =
h20d0e0x
′ = h20gPQ1 = h2
0xQ = h20Pd0x10,27 = h2
0Pd0x10,28 =h2
0Pe0B21 = h0h2d20x
′ = h0h2e0PQ1 = h0h2Pd0B21 =h0h2Pe0Q1 = h0h
23R1 = h0d0Ph2B21 = h0d0B2Pd0 =
h0e0Ph2Q1 = h0gPh2x′ = h0P
2h2B23 = h22d0PQ1 =
h22Pd0Q1 = h2
2Pe0x′ = h2d0Ph2Q1 = h2e0Ph2x
′ = h2B2P2e0 =
h2P2h2x10,27 = h2P
2h2x10,28 = h3rQ = c0ix′ = e0B2P
2h2 =Ph2
2x10,27 = Ph22x10,28 = Ph2B2Pe0 = yi2 = Pc0P
2D1 =Q2P
3h2 = B4P2c0
21 (10) d0R1
(01) d30v = d2
0e0u = d20rj = d0e0ri = d0gPv = d0kz = d0Pe0w =
e20Pv = e0gPu = e0jz = e0Pd0w = e0Pe0v = grPj = giz =
gPd0v = gPe0u = rkPe0 = rlPd0 = ijm = ikl = j2l = jk2
22 (1) h0d0R1 = h2P2R2 = Ph2PR2 = P 2h2R2 = R1P
2d0
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 83
Stem 84 continued23 (1) h2
0d0R1 = h0h2P2R2 = h0Ph2PR2 = h0P
2h2R2 = h0R1P2d0 =
h1d0x18,20 = h1x′P 2d0 = h2Ph2R1 = d0P
2h1x′ = Ph1Pd0x
′ =B1P
3d0 = B21P3h1
24 (1) d60 = d3
0e0Pe0 = d30gPd0 = d2
0e20Pd0 = d0e0gP 2e0 = d0g
2P 2d0 =d0gPe2
0 = d0iPu = e30P
2e0 = e20gP 2d0 = e2
0Pe20 = e0gPd0Pe0 =
g2Pd20 = iPd0u = jP 2v = kP 2u = vP 2j = wP 2i = PjPv
27 (1) d30P
2j = d20e0P
2i = d20iP
2e0 = d20jP
2d0 = d20Pd0Pj =
d0e0iP2d0 = d0gP 3j = d0iPd0Pe0 = d0jPd2
0 = d0kP 3e0 =d0lP
3d0 = e20P
3j = e0iPd20 = e0jP
3e0 = e0kP 3d0 = e0Pe0P2j =
e0PjP 2e0 = giP 3e0 = gjP 3d0 = gPd0P2j = gPe0P
2i =gPjP 2d0 = jPe0P
2e0 = kPd0P2e0 = kPe0P
2d0 = lPd0P2d0 =
mP 4e0 = Pe20Pj
30 (1) rP 5d0 = i2P 3d0 = iPd0P2i = jP 4j = kP 4i = PjP 3j = P 2j2
36 (1) d0P7d0 = Pd0P
6d0 = P 2d0P5d0 = P 3d0P
4d0
37 (1) h0d0P7d0 = h0Pd0P
6d0 = h0P2d0P
5d0 = h0P3d0P
4d0 =h2P
8e0 = e0P8h2 = Ph2P
7e0 = Pe0P7h2 = P 2h2P
6e0 =P 2e0P
6h2 = P 3h2P5e0 = P 3e0P
5h2 = P 4h2P4e0
38 (1) h20d0P
7d0 = h20Pd0P
6d0 = h20P
2d0P5d0 = h2
0P3d0P
4d0 =h0h2P
8e0 = h0e0P8h2 = h0Ph2P
7e0 = h0Pe0P7h2 =
h0P2h2P
6e0 = h0P2e0P
6h2 = h0P3h2P
5e0 = h0P3e0P
5h2 =h0P
4h2P4e0 = h2
2P8d0 = h2d0P
8h2 = h2Ph2P7d0 =
h2Pd0P7h2 = h2P
2h2P6d0 = h2P
2d0P6h2 = h2P
3h2P5d0 =
h2P3d0P
5h2 = h2P4h2P
4d0 = d0Ph2P7h2 = d0P
2h2P6h2 =
d0P3h2P
5h2 = d0P4h2
2 = Ph22P
6d0 = Ph2Pd0P6h2 =
Ph2P2h2P
5d0 = Ph2P2d0P
5h2 = Ph2P3h2P
4d0 =Ph2P
3d0P4h2 = Pd0P
2h2P5h2 = Pd0P
3h2P4h2 = P 2h2
2P4d0 =
P 2h2P2d0P
4h2 = P 2h2P3h2P
3d0 = P 2d0P3h2
2
84 ROBERT R. BRUNER
Stem 853 (1) c3
4 (1) h0c3
5 (010) h1f2 = h4p1
(110) h2h6c1
(001) h20c3
6 (10) x6,68
(01) h30c3
7 (1) h40c3 = h0x6,68 = h2h4Q3 = h3t1 = c2g2
8 (1) h50c3 = h2
0x6,68 = h0h2h4Q3 = h0h3t1 = h0c2g2 = h23x6,47 = e0d2
9 (1) h6Pd0
10 (1) h0h6Pd0
11 (10) h1Px6,53 = h5R1 = Ph1x6,53
(01) h20h6Pd0 = h2h6P
2h2 = h6Ph22
13 (1) x13,46
14 (10) gB23
(01) h0x13,46
15 (10) h1x14,46 = h22x13,42 = f0x11,35 = nR1
(01) h20x13,46
16 (100) x16,42
(001) h30x13,46
(011) h1x15,42 = h1x15,43 = qx′ = B1u
17 (100) g2w = grm
(010) d0x13,35 = ix10,28 = jB21 = kQ1 = lx′ = Pe0B4
(001) h40x13,46 = h2x16,38 = d0x13,34
(011) e0P2D1 = gR2 = ix10,27
18 (10) h0d0x13,35 = h0ix10,28 = h0jB21 = h0kQ1 = h0lx′ = h0Pe0B4 =
h2iB21 = h2jQ1 = h2kx′ = h2Pd0B4 = h4R1 = d0f0x′ =
d0Ph2B4 = d0iB2
(01) h50x13,46 = h0h2x16,38 = h0d0x13,34 = h2
2x16,35
(11) h0e0P2D1 = h0gR2 = h0ix10,27 = h2d0P
2D1 = h2e0R2 = d0e0R1
19 (1) h20d0x13,35 = h2
0e0P2D1 = h2
0gR2 = h20ix10,27 = h2
0ix10,28 =h2
0jB21 = h20kQ1 = h2
0lx′ = h2
0Pe0B4 = h0h2d0P2D1 =
h0h2e0R2 = h0h2iB21 = h0h2jQ1 = h0h2kx′ = h0h2Pd0B4 =h0h4R1 = h0d0e0R1 = h0d0f0x
′ = h0d0Ph2B4 = h0d0iB2 =h1d
20Q1 = h1d0e0x
′ = h1gPQ1 = h1Pd0x10,27 = h1Pd0x10,28 =h1Pe0B21 = h2
2d0R2 = h22iQ1 = h2
2jx′ = h2d
20R1 =
h2f0PQ1 = h2Ph2x13,35 = h2B2Pj = c0x16,33 = d0Ph1x10,27 =d0Ph1x10,28 = d0B1Pe0 = e0Ph1B21 = e0B1Pd0 = f0Ph2Q1 =gPh1Q1 = gPh2R1 = Ph2jB2 = D2P
3h2 = P 2h2B5 =P 2h2PD2
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 85
Stem 85 continued20 (1) d2
0e0g2 = d0e
30g = d0rz = d0jw = d0kv = d0lu = e5
0 = e0iw =e0jv = e0ku = g3Pe0 = giv = gju = r2Pe0 = ril = rjk = mPv
21 (1) Ph1x16,32
23 (1) d40k = d3
0e0j = d30gi = d2
0e20i = d2
0lP e0 = d20mPd0 = d0e0gPj =
d0e0kPe0 = d0e0lPd0 = d0gjPe0 = d0gkPd0 = e30Pj = e2
0jPe0 =e20kPd0 = e0giPe0 = e0gjPd0 = e0mP 2e0 = g2iPd0 = glP 2e0 =
gmP 2d0 = rP 2u = i2u = mPe20
26 (10) P 4x′
(01) d20iP j = d0rP
3e0 = d0i2Pe0 = d0ijPd0 = d0kP 2j = d0lP
2i =d0zP 2d0 = e0rP
3d0 = e0i2Pd0 = e0jP
2j = e0kP 2i = e0Pj2 =giP 2j = gjP 2i = rPd0P
2e0 = rPe0P2d0 = ikP 2e0 = ilP 2d0 =
j2P 2e0 = jkP 2d0 = jPe0Pj = kPd0Pj = mP 3j = Pd20z
27 (1) h0P4x′
28 (1) h20P
4x′
29 (10) d0P4u = Pd0P
3u = uP 4d0 = P 2d0P2u = PuP 3d0
(01) h30P
4x′
(11) rP 4i = i2P 2i
30 (1) h40P
4x′ = h0rP4i = h0i
2P 2i
31 (1) h50P
4x′ = h20rP
4i = h20i
2P 2i = h3iP4i = h3P
2i2
32 (01) h60P
4x′ = h30rP
4i = h30i
2P 2i = h0h3iP4i = h0h3P
2i2
(11) d20P
5e0 = d0e0P5d0 = d0Pd0P
4e0 = d0Pe0P4d0 = d0P
2d0P3e0 =
d0P2e0P
3d0 = e0Pd0P4d0 = e0P
2d0P3d0 = gP 6e0 = Pd2
0P3e0 =
Pd0Pe0P3d0 = Pd0P
2d0P2e0 = Pe0P
2d20
33 (1) h70P
4x′ = h40rP
4i = h40i
2P 2i = h20h3iP
4i = h20h3P
2i2 =h0d
20P
5e0 = h0d0e0P5d0 = h0d0Pd0P
4e0 = h0d0Pe0P4d0 =
h0d0P2d0P
3e0 = h0d0P2e0P
3d0 = h0e0Pd0P4d0 =
h0e0P2d0P
3d0 = h0gP 6e0 = h0Pd20P
3e0 = h0Pd0Pe0P3d0 =
h0Pd0P2d0P
2e0 = h0Pe0P2d2
0 = h2d20P
5d0 = h2d0Pd0P4d0 =
h2d0P2d0P
3d0 = h2e0P6e0 = h2gP 6d0 = h2Pd2
0P3d0 =
h2Pd0P2d2
0 = h2Pe0P5e0 = h2P
2e0P4e0 = h2P
3e20 = h2
3P6i =
h4P7d0 = d3
0P5h2 = d2
0Ph2P4d0 = d2
0Pd0P4h2 = d2
0P2h2P
3d0 =d20P
2d0P3h2 = d0gP 6h2 = d0Ph2Pd0P
3d0 = d0Ph2P2d2
0 =d0Pd2
0P3h2 = d0Pd0P
2h2P2d0 = e2
0P6h2 = e0Ph2P
5e0 =e0Pe0P
5h2 = e0P2h2P
4e0 = e0P2e0P
4h2 = e0P3h2P
3e0 =gPh2P
5d0 = gPd0P5h2 = gP 2h2P
4d0 = gP 2d0P4h2 =
gP 3h2P3d0 = Ph2Pd2
0P2d0 = Ph2Pe0P
4e0 = Ph2P2e0P
3e0 =Pd3
0P2h2 = Pe2
0P4h2 = Pe0P
2h2P3e0 = Pe0P
2e0P3h2 =
P 2h2P2e2
0
continued
86 ROBERT R. BRUNER
Stem 85 continued34 (1) h8
0P4x′ = h5
0rP4i = h5
0i2P 2i = h3
0h3iP4i = h3
0h3P2i2 =
h20d
20P
5e0 = h20d0e0P
5d0 = h20d0Pd0P
4e0 = h20d0Pe0P
4d0 =h2
0d0P2d0P
3e0 = h20d0P
2e0P3d0 = h2
0e0Pd0P4d0 =
h20e0P
2d0P3d0 = h2
0gP 6e0 = h20Pd2
0P3e0 = h2
0Pd0Pe0P3d0 =
h20Pd0P
2d0P2e0 = h2
0Pe0P2d2
0 = h0h2d20P
5d0 =h0h2d0Pd0P
4d0 = h0h2d0P2d0P
3d0 = h0h2e0P6e0 =
h0h2gP 6d0 = h0h2Pd20P
3d0 = h0h2Pd0P2d2
0 = h0h2Pe0P5e0 =
h0h2P2e0P
4e0 = h0h2P3e2
0 = h0h23P
6i = h0h4P7d0 =
h0d30P
5h2 = h0d20Ph2P
4d0 = h0d20Pd0P
4h2 = h0d20P
2h2P3d0 =
h0d20P
2d0P3h2 = h0d0gP 6h2 = h0d0Ph2Pd0P
3d0 =h0d0Ph2P
2d20 = h0d0Pd2
0P3h2 = h0d0Pd0P
2h2P2d0 =
h0e20P
6h2 = h0e0Ph2P5e0 = h0e0Pe0P
5h2 = h0e0P2h2P
4e0 =h0e0P
2e0P4h2 = h0e0P
3h2P3e0 = h0gPh2P
5d0 =h0gPd0P
5h2 = h0gP 2h2P4d0 = h0gP 2d0P
4h2 = h0gP 3h2P3d0 =
h0Ph2Pd20P
2d0 = h0Ph2Pe0P4e0 = h0Ph2P
2e0P3e0 =
h0Pd30P
2h2 = h0Pe20P
4h2 = h0Pe0P2h2P
3e0 =h0Pe0P
2e0P3h2 = h0P
2h2P2e2
0 = h22d0P
6e0 = h22e0P
6d0 =h2
2Pd0P5e0 = h2
2Pe0P5d0 = h2
2P2d0P
4e0 = h22P
2e0P4d0 =
h22P
3d0P3e0 = h2d0e0P
6h2 = h2d0Ph2P5e0 = h2d0Pe0P
5h2 =h2d0P
2h2P4e0 = h2d0P
2e0P4h2 = h2d0P
3h2P3e0 =
h2e0Ph2P5d0 = h2e0Pd0P
5h2 = h2e0P2h2P
4d0 =h2e0P
2d0P4h2 = h2e0P
3h2P3d0 = h2Ph2Pd0P
4e0 =h2Ph2Pe0P
4d0 = h2Ph2P2d0P
3e0 = h2Ph2P2e0P
3d0 =h2Pd0Pe0P
4h2 = h2Pd0P2h2P
3e0 = h2Pd0P2e0P
3h2 =h2Pe0P
2h2P3d0 = h2Pe0P
2d0P3h2 = h2P
2h2P2d0P
2e0 =c0iP
5d0 = c0Pd0P4i = c0P
2iP 3d0 = d0e0Ph2P5h2 =
d0e0P2h2P
4h2 = d0e0P3h2
2 = d0Ph22P
4e0 = d0Ph2Pe0P4h2 =
d0Ph2P2h2P
3e0 = d0Ph2P2e0P
3h2 = d0Pc0P4i = d0iP
5c0 =d0Pe0P
2h2P3h2 = d0P
2h22P
2e0 = d0P3c0P
2i = e0Ph22P
4d0 =e0Ph2Pd0P
4h2 = e0Ph2P2h2P
3d0 = e0Ph2P2d0P
3h2 =e0Pd0P
2h2P3h2 = e0P
2h22P
2d0 = Ph22Pd0P
3e0 =Ph2
2Pe0P3d0 = Ph2
2P2d0P
2e0 = Ph2Pd0Pe0P3h2 =
Ph2Pd0P2h2P
2e0 = Ph2Pe0P2h2P
2d0 = Pc0iP4d0 =
Pc0P2d0P
2i = iPd0P4c0 = iP 2c0P
3d0 = iP 2d0P3c0 = kP 6c0 =
Pd0Pe0P2h2
2 = Pd0P2c0P
2i
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 87
Stem 85 continued35 (1) d0P
6i = iP 6d0 = P 2d0P4i = P 2iP 4d0
36 (1) h0d0P6i = h0iP
6d0 = h0P2d0P
4i = h0P2iP 4d0 = h2P
7j =Ph2P
6j = jP 7h2 = P 2h2P5j = PjP 6h2 = P 3h2P
4j =P 2jP 5h2 = P 4h2P
3j
37 (1) h20d0P
6i = h20iP
6d0 = h20P
2d0P4i = h2
0P2iP 4d0 =
h0h2P7j = h0Ph2P
6j = h0jP7h2 = h0P
2h2P5j =
h0PjP 6h2 = h0P3h2P
4j = h0P2jP 5h2 = h0P
4h2P3j =
h1d0P7d0 = h1Pd0P
6d0 = h1P2d0P
5d0 = h1P3d0P
4d0 =h2Ph2P
6i = h2iP7h2 = h2P
3h2P4i = h2P
2iP 5h2 =d20P
7h1 = d0Ph1P6d0 = d0Pd0P
6h1 = d0P2h1P
5d0 =d0P
2d0P5h1 = d0P
3h1P4d0 = d0P
3d0P4h1 = f0P
8h2 = gP 8h1 =Ph1Pd0P
5d0 = Ph1P2d0P
4d0 = Ph1P3d2
0 = Ph2iP6h2 =
Ph2P2h2P
4i = Ph2P2iP 4h2 = iP 2h2P
5h2 = iP 3h2P4h2 =
Pd20P
5h1 = Pd0P2h1P
4d0 = Pd0P2d0P
4h1 = Pd0P3h1P
3d0 =P 2h1P
2d0P3d0 = P 2h2P
3h2P2i = P 2d2
0P3h1
88 ROBERT R. BRUNER
Stem 864 (1) h1c3
5 (1) h4h6c0
6 (10) h2h6g
(01) h21f2 = h1h4p1 = h2
2e2 = h2h4d2
7 (10) h1x6,68 = h4x6,47
(01) h0h2h6g = h22h6e0
8 (1) h6i
9 (1) h0h6i
10 (10) h1h6Pd0 = h6d0Ph1
(01) h20h6i
11 (10) gx7,40
(01) h30h6i
12 (10) x12,48
(01) h40h6i
13 (1) h50h6i
14 (10) P 3h25
(01) h1x13,46 = h22x12,44 = h3x13,42 = h4x13,34 = c1x11,35 = gB5 =
d1R1
15 (1) h0P3h2
5
16 (100) g2N = gnm = grt
(010) rQ1 = jB4
(001) h20P
3h25 = h2x15,41
17 (10) h1x16,42 = f0P2D1 = lR1
(01) h30P
3h25 = h0h2x15,41 = e0W1
(11) h0rQ1 = h0jB4 = h21x15,42 = h2
1x15,43 = h1qx′ = h1B1u =
h2rx′ = h2iB4 = h3x16,35 = D3P
3h1 = Ph1P2D3 = Pe0X1 =
PD3P2h1
19 (1) d0e0gm = d0g2l = d0rv = e3
0m = e20gl = e0g
2k = e0ru = g3j =r2j = wz
22 (100) d0P2Q1 = Pd0PQ1 = Q1 P 2d0
(010) uPu
(110) e0x18,20
(001) h1Ph1x16,32 = P 2h1W1 = uQ = X1P3h1
(101) x′P 2e0
(011) d40r = d2
0e0z = d20im = d2
0jl = d20k
2 = d0e0rPe0 = d0e0il =d0e0jk = d0grPd0 = d0gik = d0gj2 = e2
0rPd0 = e20ik = e2
0j2 =
e0gij = e0mPj = g2i2 = glP j = gPe0z = jmPe0 = klPe0 =kmPd0 = l2Pd0
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 89
Stem 86 continued23 (1) h0d0P
2Q1 = h0e0x18,20 = h0Pd0PQ1 = h0x′P 2e0 =
h0Q1 P 2d0 = h2d0x18,20 = h2x′P 2d0 = d0P
2h2x′ = Ph2Pd0x
′ =B2P
3d0 = B21P3h2
24 (1) h20d0P
2Q1 = h20e0x18,20 = h2
0Pd0PQ1 = h20x
′P 2e0 =h2
0Q1 P 2d0 = h0h2d0x18,20 = h0h2x′P 2d0 = h0d0P
2h2x′ =
h0Ph2Pd0x′ = h0B2P
3d0 = h0B21P3h2 = h2
2P3Q1 =
h2Ph2P2Q1 = h2P
2h2PQ1 = h2Q1 P 3h2 = Ph22PQ1 =
Ph2P2h2Q1 = Pc0R1
25 (10) x25,24
(01) d20P
2v = d0e0P2u = d0rP
2j = d0i2j = d0Pd0Pv = d0Pe0Pu =
d0uP 2e0 = d0vP 2d0 = e0rP2i = e0i
3 = e0Pd0Pu = e0uP 2d0 =gP 3v = riP 2e0 = rjP 2d0 = rPd0Pj = ikPj = iPd0z = j2Pj =Pd2
0v = Pd0Pe0u = wP 3e0
26 (1) h0x25,24
27 (01) h20x25,24
(11) h1P4x′ = P 2h1x18,20 = x′P 4h1
28 (10) d40P
2d0 = d30Pd2
0 = d20e0P
3e0 = d20gP 3d0 = d2
0Pe0P2e0 =
d0e20P
3d0 = d0e0Pd0P2e0 = d0e0Pe0P
2d0 = d0gPd0P2d0 =
d0Pd0Pe20 = e2
0Pd0P2d0 = e0gP 4e0 = e0Pd2
0Pe0 = g2P 4d0 =gPd3
0 = gPe0P3e0 = gP 2e2
0 = iP 3u = PuP 2i
(01) h30x25,24 = QP 2i
29 (1) h40x25,24 = h0QP 2i
30 (1) h50x25,24 = h2
0QP 2i = h3P4Q
31 (01) h60x25,24 = h3
0QP 2i = h0h3P4Q
(11) d20P
4j = d0e0P4i = d0iP
4e0 = d0jP4d0 = d0Pd0P
3j =d0PjP 3d0 = d0P
2d0P2j = d0P
2e0P2i = e0iP
4d0 = e0P2d0P
2i =gP 5j = iPd0P
3e0 = iPe0P3d0 = iP 2d0P
2e0 = jPd0P3d0 =
jP 2d20 = kP 5e0 = lP 5d0 = Pd2
0P2j = Pd0Pe0P
2i = Pd0PjP 2d0
32 (1) h70x25,24 = h4
0QP 2i = h20h3P
4Q = h0d20P
4j = h0d0e0P4i =
h0d0iP4e0 = h0d0jP
4d0 = h0d0Pd0P3j = h0d0PjP 3d0 =
h0d0P2d0P
2j = h0d0P2e0P
2i = h0e0iP4d0 = h0e0P
2d0P2i =
h0gP 5j = h0iPd0P3e0 = h0iPe0P
3d0 = h0iP2d0P
2e0 =h0jPd0P
3d0 = h0jP2d2
0 = h0kP 5e0 = h0lP5d0 = h0Pd2
0P2j =
h0Pd0Pe0P2i = h0Pd0PjP 2d0 = h2d
20P
4i = h2d0iP4d0 =
h2d0P2d0P
2i = h2e0P5j = h2iPd0P
3d0 = h2iP2d2
0 =h2jP
5e0 = h2kP 5d0 = h2Pd20P
2i = h2Pe0P4j = h2PjP 4e0 =
h2P2e0P
3j = h2P2jP 3e0 = d2
0iP4h2 = d2
0P2h2P
2i = d0f0P5d0 =
d0Ph2iP3d0 = d0Ph2Pd0P
2i = d0iPd0P3h2 = d0iP
2h2P2d0 =
d0kP 5h2 = e0Ph2P4j = e0jP
5h2 = e0P2h2P
3j = e0PjP 4h2 =(continued)
90 ROBERT R. BRUNER
Stem 86 continued32 (1) (continued) = e0P
3h2P2j = f0Pd0P
4d0 = f0P2d0P
3d0 =gPh2P
4i = giP 5h2 = gP 3h2P2i = Ph2iPd0P
2d0 = Ph2jP4e0 =
Ph2kP 4d0 = Ph2Pe0P3j = Ph2PjP 3e0 = Ph2P
2e0P2j =
iPd20P
2h2 = jPe0P4h2 = jP 2h2P
3e0 = jP 2e0P3h2 =
kPd0P4h2 = kP 2h2P
3d0 = kP 2d0P3h2 = mP 6h2 =
Pe0P2h2P
2j = Pe0PjP 3h2 = P 2h2PjP 2e0
33 (1) h80x25,24 = h5
0QP 2i = h30h3P
4Q = h20d
20P
4j = h20d0e0P
4i =h2
0d0iP4e0 = h2
0d0jP4d0 = h2
0d0Pd0P3j = h2
0d0PjP 3d0 =h2
0d0P2d0P
2j = h20d0P
2e0P2i = h2
0e0iP4d0 = h2
0e0P2d0P
2i =h2
0gP 5j = h20iPd0P
3e0 = h20iPe0P
3d0 = h20iP
2d0P2e0 =
h20jPd0P
3d0 = h20jP
2d20 = h2
0kP 5e0 = h20lP
5d0 =h2
0Pd20P
2j = h20Pd0Pe0P
2i = h20Pd0PjP 2d0 = h0h2d
20P
4i =h0h2d0iP
4d0 = h0h2d0P2d0P
2i = h0h2e0P5j = h0h2iPd0P
3d0 =h0h2iP
2d20 = h0h2jP
5e0 = h0h2kP 5d0 = h0h2Pd20P
2i =h0h2Pe0P
4j = h0h2PjP 4e0 = h0h2P2e0P
3j = h0h2P2jP 3e0 =
h0d20iP
4h2 = h0d20P
2h2P2i = h0d0f0P
5d0 = h0d0Ph2iP3d0 =
h0d0Ph2Pd0P2i = h0d0iPd0P
3h2 = h0d0iP2h2P
2d0 =h0d0kP 5h2 = h0e0Ph2P
4j = h0e0jP5h2 = h0e0P
2h2P3j =
h0e0PjP 4h2 = h0e0P3h2P
2j = h0f0Pd0P4d0 = h0f0P
2d0P3d0 =
h0gPh2P4i = h0giP 5h2 = h0gP 3h2P
2i = h0Ph2iPd0P2d0 =
h0Ph2jP4e0 = h0Ph2kP 4d0 = h0Ph2Pe0P
3j = h0Ph2PjP 3e0 =h0Ph2P
2e0P2j = h0iPd2
0P2h2 = h0jPe0P
4h2 = h0jP2h2P
3e0 =h0jP
2e0P3h2 = h0kPd0P
4h2 = h0kP 2h2P3d0 = h0kP 2d0P
3h2 =h0mP 6h2 = h0Pe0P
2h2P2j = h0Pe0PjP 3h2 = h0P
2h2PjP 2e0 =h1d
20P
5e0 = h1d0e0P5d0 = h1d0Pd0P
4e0 = h1d0Pe0P4d0 =
h1d0P2d0P
3e0 = h1d0P2e0P
3d0 = h1e0Pd0P4d0 =
h1e0P2d0P
3d0 = h1gP 6e0 = h1Pd20P
3e0 = h1Pd0Pe0P3d0 =
h1Pd0P2d0P
2e0 = h1Pe0P2d2
0 = h22d0P
5j = h22iP
5e0 =h2
2jP5d0 = h2
2Pd0P4j = h2
2Pe0P4i = h2
2PjP 4d0 =h2
2P2d0P
3j = h22P
2iP 3e0 = h22P
2jP 3d0 = h2d0Ph2P4j =
h2d0jP5h2 = h2d0P
2h2P3j = h2d0PjP 4h2 = h2d0P
3h2P2j =
h2e0Ph2P4i = h2e0iP
5h2 = h2e0P3h2P
2i = h2f0P6e0 =
h2Ph2iP4e0 = h2Ph2jP
4d0 = h2Ph2Pd0P3j = h2Ph2PjP 3d0 =
h2Ph2P2d0P
2j = h2Ph2P2e0P
2i = h2iPe0P4h2 =
h2iP2h2P
3e0 = h2iP2e0P
3h2 = h2jPd0P4h2 = h2jP
2h2P3d0 =
h2jP2d0P
3h2 = h2lP6h2 = h2Pd0P
2h2P2j = h2Pd0PjP 3h2 =
h2Pe0P2h2P
2i = h2P2h2PjP 2d0 = c0iP
4i = c0P2i2 =
d20e0P
5h1 = d20Ph1P
4e0 = d20Pe0P
4h1 = d20P
2h1P3e0 =
d20P
2e0P3h1 = d0e0Ph1P
4d0 = d0e0Pd0P4h1 = d0e0P
2h1P3d0 =
d0e0P2d0P
3h1 = d0Ph1Pd0P3e0 = (continued)
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 91
Stem 86 continued33 (1) (continued) = d0Ph1Pe0P
3d0 = d0Ph1P2d0P
2e0 =d0Ph2
2P3j = d0Ph2jP
4h2 = d0Ph2P2h2P
2j = d0Ph2PjP 3h2 =d0jP
2h2P3h2 = d0Pd0Pe0P
3h1 = d0Pd0P2h1P
2e0 =d0Pe0P
2h1P2d0 = d0P
2h22Pj = e0f0P
6h2 = e0gP 6h1 =e0Ph1Pd0P
3d0 = e0Ph1P2d2
0 = e0Ph2iP4h2 = e0Ph2P
2h2P2i =
e0iP2h2P
3h2 = e0Pd20P
3h1 = e0Pd0P2h1P
2d0 = f0Ph2P5e0 =
f0Pe0P5h2 = f0P
2h2P4e0 = f0P
2e0P4h2 = f0P
3h2P3e0 =
gPh1P5e0 = gPe0P
5h1 = gP 2h1P4e0 = gP 2e0P
4h1 =gP 3h1P
3e0 = Ph1Pd20P
2e0 = Ph1Pd0Pe0P2d0 = Ph2
2iP3e0 =
Ph22jP
3d0 = Ph22Pd0P
2j = Ph22Pe0P
2i = Ph22PjP 2d0 =
Ph2iPe0P3h2 = Ph2iP
2h2P2e0 = Ph2jPd0P
3h2 =Ph2jP
2h2P2d0 = Ph2lP
5h2 = Ph2Pd0P2h2Pj = rP 6c0 =
i2P 4c0 = iPe0P2h2
2 = iP 2c0P2i = jPd0P
2h22 = lP 2h2P
4h2 =lP 3h2
2 = Pd20Pe0P
2h1
92 ROBERT R. BRUNER
Stem 875 (1) h2
1c3 = h2f2 = h3e2
6 (1) h1h4h6c0 = h3h6e0
7 (1) x7,74
9 (1) h24Q2 = gQ3
10 (1) x10,60
11 (1) h21h6Pd0 = h1h6d0Ph1 = h3h6P
2h1 = h6c0Pc0
12 (1) h6P2c0
13 (10) gx9,39 = rQ2
(01) h1x12,48 = h3x12,44
15 (10) x15,47
(01) h1P3h2
5 = h2x14,46 = h3x14,42 = gx11,35
17 (1) x17,50
18 (10) e0g2r = e0m
2 = glm = vw
(01) d20B21 = d0e0Q1 = d0gx′ = e2
0x′ = Pd0B23 = Pe0x10,27 =
Pe0x10,28
19 (1) h0d20B21 = h0d0e0Q1 = h0d0gx′ = h0e
20x
′ = h0Pd0B23 =h0Pe0x10,27 = h0Pe0x10,28 = h2d
20Q1 = h2d0e0x
′ = h2gPQ1 =h2Pd0x10,27 = h2Pd0x10,28 = h2Pe0B21 = h4P
2Q1 =c0x16,35 = d0Ph2x10,27 = d0Ph2x10,28 = d0B2Pe0 = e0Ph2B21 =e0B2Pd0 = gPh2Q1 = Q2P
2d0
20 (1) Px16,35
21 (100) d0PR2 = e0R1 = Pd0R2
(010) d30w = d2
0e0v = d20gu = d2
0rk = d0e20u = d0e0rj = d0gri = d0lz =
e20ri = e0gPv = e0kz = e0Pe0w = g2Pu = gjz = gPd0w =
gPe0v = rlPe0 = rmPd0 = ikm = il2 = j2m = jkl = k3
(001) h0Px16,35 = Ph2x16,32
(101) iPQ1 = x′Pj
22 (10) h0iPQ1 = h0x′Pj = f0x18,20 = Ph2ix
′ = B2P2i = B4P
3h2
(01) h20Px16,35 = h0Ph2x16,32
(11) h0d0PR2 = h0e0R1 = h0Pd0R2 = h2d0R1 = P 2h2P2D1 =
R1P2e0
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 93
Stem 87 continued23 (10) h2
0iPQ1 = h20x
′Pj = h0f0x18,20 = h0Ph2ix′ = h0B2P
2i =h0B4P
3h2 = h1e0x18,20 = h1x′P 2e0 = e0P
2h1x′ = Ph1Pe0x
′ =B1P
3e0 = x10,28P3h1
(01) h30Px16,35 = h2
0Ph2x16,32 = h21Ph1x16,32 = h1P
2h1W1 = h1uQ =h1uPu = h1X1P
3h1 = h3P3Q1 = Ph2
1W1 = Ph1P2h1X1 =
Ph1u2 = qP 2u = GP 4h1
(11) h20d0PR2 = h2
0e0R1 = h20Pd0R2 = h0h2d0R1 = h0P
2h2P2D1 =
h0R1P2e0 = h1d0P
2Q1 = h1Pd0PQ1 = h1Q1 P 2d0 = h22P
2R2 =h2Ph2PR2 = h2P
2h2R2 = h2R1P2d0 = d0Ph1PQ1 =
d0P2h1Q1 = d0P
2h2R1 = Ph1Pd0Q1 = Ph22R2 = Ph2Pd0R1 =
x10,27P3h1
24 (1) d50e0 = d3
0gPe0 = d20e
20Pe0 = d2
0e0gPd0 = d0e30Pd0 = d0g
2P 2e0 =d0iPv = d0jPu = d0uPj = e2
0gP 2e0 = e0g2P 2d0 = e0gPe2
0 =e0iPu = g2Pd0Pe0 = riP j = i2z = iPd0v = iPe0u = jPd0u =kP 2v = lP 2u = wP 2j
27 (1) d30iPd0 = d2
0e0P2j = d2
0gP 2i = d20jP
2e0 = d20kP 2d0 = d2
0Pe0Pj =d0e
20P
2i = d0e0iP2e0 = d0e0jP
2d0 = d0e0Pd0Pj = d0giP 2d0 =d0iPe2
0 = d0jPd0Pe0 = d0kPd20 = d0lP
3e0 = d0mP 3d0 =e20iP
2d0 = e0gP 3j = e0iPd0Pe0 = e0jPd20 = e0kP 3e0 =
e0lP3d0 = giPd2
0 = gjP 3e0 = gkP 3d0 = gPe0P2j = gPjP 2e0 =
kPe0P2e0 = lPd0P
2e0 = lP e0P2d0 = mPd0P
2d0
28 (1) h21P
4x′ = h1P2h1x18,20 = h1x
′P 4h1 = Ph21x18,20 = Ph1x
′P 3h1 =B1P
5h1 = P 2h21x
′
30 (1) d0iP3j = d0PjP 2i = rP 5e0 = i2P 3e0 = ijP 3d0 = iPd0P
2j =iPe0P
2i = iP jP 2d0 = jPd0P2i = kP 4j = lP 4i = zP 4d0
33 (1) P 6u
36 (1) d0P7e0 = e0P
7d0 = Pd0P6e0 = Pe0P
6d0 = P 2d0P5e0 =
P 2e0P5d0 = P 3d0P
4e0 = P 3e0P4d0
37 (1) h0d0P7e0 = h0e0P
7d0 = h0Pd0P6e0 = h0Pe0P
6d0 =h0P
2d0P5e0 = h0P
2e0P5d0 = h0P
3d0P4e0 = h0P
3e0P4d0 =
h2d0P7d0 = h2Pd0P
6d0 = h2P2d0P
5d0 = h2P3d0P
4d0 =d20P
7h2 = d0Ph2P6d0 = d0Pd0P
6h2 = d0P2h2P
5d0 =d0P
2d0P5h2 = d0P
3h2P4d0 = d0P
3d0P4h2 = gP 8h2 =
Ph2Pd0P5d0 = Ph2P
2d0P4d0 = Ph2P
3d20 = Pd2
0P5h2 =
Pd0P2h2P
4d0 = Pd0P2d0P
4h2 = Pd0P3h2P
3d0 =P 2h2P
2d0P3d0 = P 2d2
0P3h2
continued
94 ROBERT R. BRUNER
Stem 87 continued38 (1) h2
0d0P7e0 = h2
0e0P7d0 = h2
0Pd0P6e0 = h2
0Pe0P6d0 =
h20P
2d0P5e0 = h2
0P2e0P
5d0 = h20P
3d0P4e0 = h2
0P3e0P
4d0 =h0h2d0P
7d0 = h0h2Pd0P6d0 = h0h2P
2d0P5d0 =
h0h2P3d0P
4d0 = h0d20P
7h2 = h0d0Ph2P6d0 = h0d0Pd0P
6h2 =h0d0P
2h2P5d0 = h0d0P
2d0P5h2 = h0d0P
3h2P4d0 =
h0d0P3d0P
4h2 = h0gP 8h2 = h0Ph2Pd0P5d0 =
h0Ph2P2d0P
4d0 = h0Ph2P3d2
0 = h0Pd20P
5h2 =h0Pd0P
2h2P4d0 = h0Pd0P
2d0P4h2 = h0Pd0P
3h2P3d0 =
h0P2h2P
2d0P3d0 = h0P
2d20P
3h2 = h22P
8e0 = h2e0P8h2 =
h2Ph2P7e0 = h2Pe0P
7h2 = h2P2h2P
6e0 = h2P2e0P
6h2 =h2P
3h2P5e0 = h2P
3e0P5h2 = h2P
4h2P4e0 = e0Ph2P
7h2 =e0P
2h2P6h2 = e0P
3h2P5h2 = e0P
4h22 = Ph2
2P6e0 =
Ph2Pe0P6h2 = Ph2P
2h2P5e0 = Ph2P
2e0P5h2 =
Ph2P3h2P
4e0 = Ph2P3e0P
4h2 = Pc0P6i = iP 7c0 =
Pe0P2h2P
5h2 = Pe0P3h2P
4h2 = P 2h22P
4e0 = P 2h2P2e0P
4h2 =P 2h2P
3h2P3e0 = P 2e0P
3h22 = P 3c0P
4i = P 2iP 5c0
39 (1) P 8i
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 95
Stem 888 (10) g2
2
(01) h1x7,74
(11) h24D2 = h5Q2
9 (10) h6Pe0
(01) h0h24D2 = h0h5Q2 = h0g
22
10 (1) h0h6Pe0 = h2h6Pd0 = h6d0Ph2
11 (1) h20h6Pe0 = h0h2h6Pd0 = h0h6d0Ph2
12 (100) x12,51
(001) h30h6Pe0 = h2
0h2h6Pd0 = h20h6d0Ph2 = h3
1h6Pd0 = h21h6d0Ph1 =
h1h3h6P2h1 = h1h6c0Pc0 = h2
2h6P2h2 = h2h6Ph2
2 = h3h6Ph21 =
h6c20Ph1 = d0x8,51
(011) gG21 = nQ2 = rD2
13 (10) h1h6P2c0 = h6c0P
2h1 = h6Ph1Pc0
(01) h0x12,51 = h3x12,45
14 (10) h6P3h1
(01) h20x12,51 = h0h3x12,45 = h2
1x12,48 = h1h3x12,44 = d1Q1
16 (1) x16,48
17 (1) d20B4 = e0x13,35 = gP 2D1 = iB23 = jx10,27 = jx10,28 = kB21 =
lQ1 = mx′
18 (10) h1x17,50
(01) h0d20B4 = h0e0x13,35 = h0gP 2D1 = h0iB23 = h0jx10,27 =
h0jx10,28 = h0kB21 = h0lQ1 = h0mx′ = h2d0x13,35 =h2e0P
2D1 = h2gR2 = h2ix10,27 = h2ix10,28 = h2jB21 =h2kQ1 = h2lx
′ = h2Pe0B4 = h4PR2 = d0f0Q1 = d0gR1 =d0jB2 = e2
0R1 = e0f0x′ = e0Ph2B4 = e0iB2 = D2P
2d0 =Pd0B5 = Pd0PD2
19 (1) h20d
20B4 = h2
0e0x13,35 = h20gP 2D1 = h2
0iB23 = h20jx10,27 =
h20jx10,28 = h2
0kB21 = h20lQ1 = h2
0mx′ = h0h2d0x13,35 =h0h2e0P
2D1 = h0h2gR2 = h0h2ix10,27 = h0h2ix10,28 =h0h2jB21 = h0h2kQ1 = h0h2lx
′ = h0h2Pe0B4 = h0h4PR2 =h0d0f0Q1 = h0d0gR1 = h0d0jB2 = h0e
20R1 = h0e0f0x
′ =h0e0Ph2B4 = h0e0iB2 = h0D2P
2d0 = h0Pd0B5 = h0Pd0PD2 =h1d
20B21 = h1d0e0Q1 = h1d0gx′ = h1e
20x
′ = h1Pd0B23 =h1Pe0x10,27 = h1Pe0x10,28 = h2
2d0P2D1 = h2
2e0R2 = h22iB21 =
h22jQ1 = h2
2kx′ = h22Pd0B4 = h2h4R1 = h2d0e0R1 = h2d0f0x
′ =h2d0Ph2B4 = h2d0iB2 = c0x16,37 = d3
0B1 = d0Ph1B23 =e0Ph1x10,27 = e0Ph1x10,28 = e0B1Pe0 = f0Ph2B21 = f0B2Pd0 =gPh1B21 = gB1Pd0 = Ph2kB2 = AP 3h2 = P 2h2PA
continued
96 ROBERT R. BRUNER
Stem 88 continued20 (10) iR2
(01) d20g
3 = d20r
2 = d0e20g
2 = d0kw = d0lv = d0mu = e40g = e0rz =
e0jw = e0kv = e0lu = giw = gjv = gku = rim = rjl = rk2
21 (1) h0iR2 = h1Px16,35 = f0R1 = Ph1x16,35 = Ph2x16,33 = R1Pj
23 (1) d40l = d3
0e0k = d30gj = d2
0e20j = d2
0e0gi = d20mPe0 = d0e
30i =
d0e0lP e0 = d0e0mPd0 = d0g2Pj = d0gkPe0 = d0glPd0 =
e20gPj = e2
0kPe0 = e20lPd0 = e0gjPe0 = e0gkPd0 = g2iPe0 =
g2jPd0 = gmP 2e0 = rP 2v = i2v = iju = zPu
26 (01) d30i
2 = d20rP
2d0 = d20jP j = d0e0iP j = d0rPd2
0 = d0ijPe0 =d0ikPd0 = d0j
2Pd0 = d0lP2j = d0mP 2i = d0zP 2e0 = e0rP
3e0 =e0i
2Pe0 = e0ijPd0 = e0kP 2j = e0lP2i = e0zP 2d0 = grP 3d0 =
gi2Pd0 = gjP 2j = gkP 2i = gPj2 = rPe0P2e0 = ilP 2e0 =
imP 2d0 = jkP 2e0 = jlP 2d0 = k2P 2d0 = kPe0Pj = lPd0Pj =Pd0Pe0z
(11) P 4Q1
27 (1) h0P4Q1 = h2P
4x′ = P 2h2x18,20 = x′P 4h2
28 (1) h20P
4Q1 = h0h2P4x′ = h0P
2h2x18,20 = h0x′P 4h2
29 (10) d0P4v = Pd0P
3v = vP 4d0 = P 2d0P2v = PvP 3d0
(01) h30P
4Q1 = h20h2P
4x′ = h20P
2h2x18,20 = h20x
′P 4h2 = h31P
4x′ =h2
1P2h1x18,20 = h2
1x′P 4h1 = h1Ph2
1x18,20 = h1Ph1x′P 3h1 =
h1B1P5h1 = h1P
2h21x
′ = Ph21P
2h1x′ = Ph1B1P
4h1 =B1P
2h1P3h1 = QP 3e0
(11) e0P4u = rP 4j = i2P 2j = ijP 2i = Pe0P
3u = uP 4e0 =P 2e0P
2u = PuP 3e0
32 (1) d30P
4d0 = d20Pd0P
3d0 = d20P
2d20 = d0e0P
5e0 = d0gP 5d0 =d0Pd2
0P2d0 = d0Pe0P
4e0 = d0P2e0P
3e0 = e20P
5d0 =e0Pd0P
4e0 = e0Pe0P4d0 = e0P
2d0P3e0 = e0P
2e0P3d0 =
gPd0P4d0 = gP 2d0P
3d0 = Pd40 = Pd0Pe0P
3e0 = Pd0P2e2
0 =Pe2
0P3d0 = Pe0P
2d0P2e0
34 (1) h1P6u = Ph1P
4Q = Ph1P5u = P 2h1P
4u = uP 6h1 =P 3h1P
3u = QP 5h1 = PuP 5h1 = P 4h1P2u
35 (1) d0P6j = e0P
6i = iP 6e0 = jP 6d0 = Pd0P5j = PjP 5d0 =
P 2d0P4j = P 2e0P
4i = P 2iP 4e0 = P 2jP 4d0 = P 3d0P3j
36 (1) h0d0P6j = h0e0P
6i = h0iP6e0 = h0jP
6d0 = h0Pd0P5j =
h0PjP 5d0 = h0P2d0P
4j = h0P2e0P
4i = h0P2iP 4e0 =
h0P2jP 4d0 = h0P
3d0P3j = h2d0P
6i = h2iP6d0 = h2P
2d0P4i =
h2P2iP 4d0 = d0iP
6h2 = d0P2h2P
4i = d0P2iP 4h2 =
f0P7d0 = Ph2iP
5d0 = Ph2Pd0P4i = Ph2P
2iP 3d0 =iPd0P
5h2 = iP 2h2P4d0 = iP 2d0P
4h2 = iP 3h2P3d0 = kP 7h2 =
Pd0P3h2P
2i = P 2h2P2d0P
2i
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 97
Stem 88 continued37 (1) h2
0d0P6j = h2
0e0P6i = h2
0iP6e0 = h2
0jP6d0 = h2
0Pd0P5j =
h20PjP 5d0 = h2
0P2d0P
4j = h20P
2e0P4i = h2
0P2iP 4e0 =
h20P
2jP 4d0 = h20P
3d0P3j = h0h2d0P
6i = h0h2iP6d0 =
h0h2P2d0P
4i = h0h2P2iP 4d0 = h0d0iP
6h2 = h0d0P2h2P
4i =h0d0P
2iP 4h2 = h0f0P7d0 = h0Ph2iP
5d0 = h0Ph2Pd0P4i =
h0Ph2P2iP 3d0 = h0iPd0P
5h2 = h0iP2h2P
4d0 = h0iP2d0P
4h2 =h0iP
3h2P3d0 = h0kP 7h2 = h0Pd0P
3h2P2i = h0P
2h2P2d0P
2i =h1d0P
7e0 = h1e0P7d0 = h1Pd0P
6e0 = h1Pe0P6d0 =
h1P2d0P
5e0 = h1P2e0P
5d0 = h1P3d0P
4e0 = h1P3e0P
4d0 =h2
2P7j = h2Ph2P
6j = h2jP7h2 = h2P
2h2P5j = h2PjP 6h2 =
h2P3h2P
4j = h2P2jP 5h2 = h2P
4h2P3j = d0e0P
7h1 =d0Ph1P
6e0 = d0Pe0P6h1 = d0P
2h1P5e0 = d0P
2e0P5h1 =
d0P3h1P
4e0 = d0P3e0P
4h1 = e0Ph1P6d0 = e0Pd0P
6h1 =e0P
2h1P5d0 = e0P
2d0P5h1 = e0P
3h1P4d0 = e0P
3d0P4h1 =
Ph1Pd0P5e0 = Ph1Pe0P
5d0 = Ph1P2d0P
4e0 = Ph1P2e0P
4d0 =Ph1P
3d0P3e0 = Ph2
2P5j = Ph2jP
6h2 = Ph2P2h2P
4j =Ph2PjP 5h2 = Ph2P
3h2P3j = Ph2P
2jP 4h2 = jP 2h2P5h2 =
jP 3h2P4h2 = Pd0Pe0P
5h1 = Pd0P2h1P
4e0 = Pd0P2e0P
4h1 =Pd0P
3h1P3e0 = Pe0P
2h1P4d0 = Pe0P
2d0P4h1 =
Pe0P3h1P
3d0 = P 2h1P2d0P
3e0 = P 2h1P2e0P
3d0 = P 2h22P
3j =P 2h2PjP 4h2 = P 2h2P
3h2P2j = PjP 3h2
2 = P 2d0P2e0P
3h1
98 ROBERT R. BRUNER
Stem 897 (1) h2
2h6g = h4h6Ph2 = h5D2
8 (1) h6j
9 (1) h0h6j = h2h6i
10 (1) h20h6j = h0h2h6i = h1h6Pe0 = h6e0Ph1
11 (1) x11,59
12 (1) h0x11,59
13 (1) h20x11,59 = h1x12,51 = h2x12,48 = qQ2
15 (1) h1h6P3h1 = h6Ph1P
2h1
16 (1) rB21 = kB4
18 (1) x18,50
19 (10) d0g2m = d0rw = e2
0gm = e0g2l = e0rv = g3k = gru = r2k
(01) h0x18,50 = h21x17,50 = Ph1x14,42
22 (10) d0Pd0x′ = e0P
2Q1 = Pe0PQ1 = Q1 P 2e0 = B21P2d0
(01) d30e0r = d2
0gz = d20jm = d2
0kl = d0e20z = d0e0im = d0e0jl =
d0e0k2 = d0grPe0 = d0gil = d0gjk = e2
0rPe0 = e20il = e2
0jk =e0grPd0 = e0gik = e0gj2 = g2ij = gmPj = kmPe0 = l2Pe0 =lmPd0 = uPv = vPu
(11) gx18,20
23 (1) h0d0Pd0x′ = h0e0P
2Q1 = h0gx18,20 = h0Pe0PQ1 =h0Q1 P 2e0 = h0B21P
2d0 = h2d0P2Q1 = h2e0x18,20 =
h2Pd0PQ1 = h2x′P 2e0 = h2Q1 P 2d0 = d0Ph2PQ1 =
d0P2h2Q1 = e0P
2h2x′ = Ph2Pd0Q1 = Ph2Pe0x
′ = B2P3e0 =
x10,27P3h2 = x10,28P
3h2
24 (1) h20d0Pd0x
′ = h20e0P
2Q1 = h20gx18,20 = h2
0Pe0PQ1 =h2
0Q1 P 2e0 = h20B21P
2d0 = h0h2d0P2Q1 = h0h2e0x18,20 =
h0h2Pd0PQ1 = h0h2x′P 2e0 = h0h2Q1 P 2d0 = h0d0Ph2PQ1 =
h0d0P2h2Q1 = h0e0P
2h2x′ = h0Ph2Pd0Q1 = h0Ph2Pe0x
′ =h0B2P
3e0 = h0x10,27P3h2 = h0x10,28P
3h2 = h22d0x18,20 =
h22x
′P 2d0 = h2d0P2h2x
′ = h2Ph2Pd0x′ = h2B2P
3d0 =h2B21P
3h2 = c0P2R2 = d0Ph2
2x′ = d0B2P
3h2 = Ph2B2P2d0 =
Ph2P2h2B21 = Pc0PR2 = B2Pd0P
2h2 = P 2c0R2
25 (01) d30Pu = d2
0Pd0u = d0e0P2v = d0gP 2u = d0riPd0 = d0i
2k =d0ij
2 = d0Pe0Pv = d0vP 2e0 = d0wP 2d0 = d0zPj = e20P
2u =e0rP
2j = e0i2j = e0Pd0Pv = e0Pe0Pu = e0uP 2e0 = e0vP 2d0 =
grP 2i = gi3 = gPd0Pu = guP 2d0 = rjP 2e0 = rkP 2d0 =rPe0Pj = ilP j = iPe0z = jkPj = jPd0z = Pd2
0w = Pd0Pe0v =Pe2
0u
(11) P 3R2
26 (1) h0P3R2 = h2x25,24 = P 2h2R1
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 99
Stem 89 continued27 (1) h2
0P3R2 = h0h2x25,24 = h0P
2h2R1 = h1P4Q1 = Ph1P
3Q1 =P 2h1P
2Q1 = R1P4h2 = Q1 P 4h1 = P 3h1PQ1
28 (1) d40P
2e0 = d30e0P
2d0 = d30Pd0Pe0 = d2
0e0Pd20 = d2
0gP 3e0 =d0e
20P
3e0 = d0e0gP 3d0 = d0e0Pe0P2e0 = d0gPd0P
2e0 =d0gPe0P
2d0 = d0Pe30 = e3
0P3d0 = e2
0Pd0P2e0 = e2
0Pe0P2d0 =
e0gPd0P2d0 = e0Pd0Pe2
0 = g2P 4e0 = gPd20Pe0 = iP 3v = jP 3u =
uP 3j = PjP 2u = PuP 2j = PvP 2i
31 (1) d20iP
3d0 = d20Pd0P
2i = d0e0P4j = d0gP 4i = d0iPd0P
2d0 =d0jP
4e0 = d0kP 4d0 = d0Pe0P3j = d0PjP 3e0 = d0P
2e0P2j =
e20P
4i = e0iP4e0 = e0jP
4d0 = e0Pd0P3j = e0PjP 3d0 =
e0P2d0P
2j = e0P2e0P
2i = giP 4d0 = gP 2d0P2i = iPd3
0 =iPe0P
3e0 = iP 2e20 = jPd0P
3e0 = jPe0P3d0 = jP 2d0P
2e0 =kPd0P
3d0 = kP 2d20 = lP 5e0 = mP 5d0 = Pd0Pe0P
2j =Pd0PjP 2e0 = Pe2
0P2i = Pe0PjP 2d0
34 (1) iP 5j = PjP 4i = P 2iP 3j
35 (1) h0iP5j = h0PjP 4i = h0P
2iP 3j = h21P
6u = h1Ph1P4Q =
h1Ph1P5u = h1P
2h1P4u = h1uP 6h1 = h1P
3h1P3u =
h1QP 5h1 = h1PuP 5h1 = h1P4h1P
2u = f0P6i = Ph2
1P4u =
Ph1P2h1P
3u = Ph1uP 5h1 = Ph1P3h1P
2u = Ph1QP 4h1 =Ph1PuP 4h1 = Ph2iP
4i = Ph2P2i2 = rP 7h2 = qP 7h1 =
i2P 5h2 = iP 3h2P2i = P 2h2
1P2u = P 2h1uP 4h1 = P 2h1P
3h1Q =P 2h1P
3h1Pu = uP 3h21
100 ROBERT R. BRUNER
Stem 9010 (1) x10,63
11 (1) h0x10,63 = pQ2
12 (1) x12,55
13 (1) h0x12,55
14 (10) h6P3h2
(01) h20x12,55
15 (010) h0h6P3h2
(001) h30x12,55 = xx′
(101) rB4
16 (01) h40x12,55 = h0xx′ = h0rB4 = h3x15,41 = ix9,40
(11) h20h6P
3h2 = h21h6P
3h1 = h1h6Ph1P2h1 = h6Ph3
1
17 (10) x17,52
(01) h50x12,55 = h2
0xx′ = h20rB4 = h0h3x15,41 = h0ix9,40 = h3rx
′ =h3iB4
18 (100) g3r = gm2 = w2
(001) h60x12,55 = h3
0xx′ = h30rB4 = h2
0h3x15,41 = h20ix9,40 = h0h3rx
′ =h0h3iB4 = h2
3x16,32 = g2i2 = xri = ix11,35
(101) r3
(011) d20x10,27 = d2
0x10,28 = d0e0B21 = d0gQ1 = e20Q1 = e0gx′ =
Pe0B23
19 (1) h70x12,55 = h4
0xx′ = h40rB4 = h3
0h3x15,41 = h30ix9,40 = h2
0h3rx′ =
h20h3iB4 = h0h
23x16,32 = h0d
20x10,27 = h0d
20x10,28 = h0d0e0B21 =
h0d0gQ1 = h0e20Q1 = h0e0gx′ = h0g2i
2 = h0xri = h0r3 =
h0ix11,35 = h0Pe0B23 = h1x18,50 = h2d20B21 = h2d0e0Q1 =
h2d0gx′ = h2e20x
′ = h2Pd0B23 = h2Pe0x10,27 = h2Pe0x10,28 =h2
3ix′ = h4Pd0x
′ = c0x16,38 = d30B2 = d0Ph2B23 = e0Ph2x10,27 =
e0Ph2x10,28 = e0B2Pe0 = gPh2B21 = gB2Pd0 = Q2P2e0 =
B3P2d0
20 (1) d0x16,32
21 (100) e0PR2 = gR1 = Pd0P2D1 = Pe0R2
(010) d20e0w = d2
0gv = d20rl = d0e
20v = d0e0gu = d0e0rk = d0grj =
d0mz = e30u = e2
0rj = e0gri = e0lz = g2Pv = gkz = gPe0w =rmPe0 = ilm = jkm = jl2 = k2l
(001) h0d0x16,32 = h2Px16,35 = Ph2x16,35
(101) d0ix′ = jPQ1 = B4P
2d0 = Q1 Pj
22 (10) h0d0ix′ = h0jPQ1 = h0B4P
2d0 = h0Q1 Pj = h2iPQ1 =h2x
′Pj = f0P2Q1 = Ph2iQ1 = Ph2jx
′ = B2P2j = P 2h2x13,35
(01) h20d0x16,32 = h0h2Px16,35 = h0Ph2x16,35 = h2Ph2x16,32 =
P 2h2x13,34
(11) h0e0PR2 = h0gR1 = h0Pd0P2D1 = h0Pe0R2 = h2d0PR2 =
h2e0R1 = h2Pd0R2 = d0Ph2R2 = d0Pd0R1
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 101
Stem 90 continued23 (1) h2
0d0ix′ = h2
0e0PR2 = h20gR1 = h2
0jPQ1 = h20Pd0P
2D1 =h2
0Pe0R2 = h20B4P
2d0 = h20Q1 Pj = h0h2d0PR2 = h0h2e0R1 =
h0h2iPQ1 = h0h2Pd0R2 = h0h2x′Pj = h0d0Ph2R2 =
h0d0Pd0R1 = h0f0P2Q1 = h0Ph2iQ1 = h0Ph2jx
′ = h0B2P2j =
h0P2h2x13,35 = h1d0Pd0x
′ = h1e0P2Q1 = h1gx18,20 =
h1Pe0PQ1 = h1Q1 P 2e0 = h1B21P2d0 = h2
2d0R1 = h2f0x18,20 =h2Ph2ix
′ = h2B2P2i = h2P
2h2P2D1 = h2B4P
3h2 = h2R1P2e0 =
d20Ph1x
′ = d0B1P2d0 = d0P
2h1B21 = e0Ph1PQ1 = e0P2h1Q1 =
e0P2h2R1 = f0P
2h2x′ = gP 2h1x
′ = Ph1Pd0B21 = Ph1Pe0Q1 =Ph2
2P2D1 = Ph2Pe0R1 = Ph2P
2h2B4 = iB2P2h2 = B1Pd2
0 =B23P
3h1
24 (1) d50g = d4
0e20 = d2
0e0gPe0 = d20g
2Pd0 = d20iu = d0e
30Pe0 =
d0e20gPd0 = d0ri
2 = d0jPv = d0kPu = d0vPj = e40Pd0 =
e0g2P 2e0 = e0iPv = e0jPu = e0uPj = g3P 2d0 = g2Pe2
0 =giPu = r2P 2d0 = rjPj = ijz = iPd0w = iPe0v = jPd0v =jPe0u = kPd0u = lP 2v = mP 2u
27 (1) d40Pj = d3
0iPe0 = d30jPd0 = d2
0e0iPd0 = d20gP 2j = d2
0kP 2e0 =d20lP
2d0 = d0e20P
2j = d0e0gP 2i = d0e0jP2e0 = d0e0kP 2d0 =
d0e0Pe0Pj = d0giP 2e0 = d0gjP 2d0 = d0gPd0Pj = d0jPe20 =
d0kPd0Pe0 = d0lPd20 = d0mP 3e0 = e3
0P2i = e2
0iP2e0 =
e20jP
2d0 = e20Pd0Pj = e0giP 2d0 = e0iPe2
0 = e0jPd0Pe0 =e0kPd2
0 = e0lP3e0 = e0mP 3d0 = g2P 3j = giPd0Pe0 = gjPd2
0 =gkP 3e0 = glP 3d0 = lP e0P
2e0 = mPd0P2e0 = mPe0P
2d0
30 (1) d20iP
2i = d0rP4d0 = d0i
2P 2d0 = d0jP3j = d0PjP 2j = e0iP
3j =e0PjP 2i = rPd0P
3d0 = rP 2d20 = i2Pd2
0 = ijP 3e0 = ikP 3d0 =iPe0P
2j = iP jP 2e0 = j2P 3d0 = jPd0P2j = jPe0P
2i =jP jP 2d0 = kPd0P
2i = lP 4j = mP 4i = Pd0Pj2 = zP 4e0
33 (1) P 6v
36 (1) d20P
6d0 = d0Pd0P5d0 = d0P
2d0P4d0 = d0P
3d20 = e0P
7e0 =gP 7d0 = Pd2
0P4d0 = Pd0P
2d0P3d0 = Pe0P
6e0 = P 2d30 =
P 2e0P5e0 = P 3e0P
4e0
continued
102 ROBERT R. BRUNER
Stem 90 continued37 (1) h0d
20P
6d0 = h0d0Pd0P5d0 = h0d0P
2d0P4d0 = h0d0P
3d20 =
h0e0P7e0 = h0gP 7d0 = h0Pd2
0P4d0 = h0Pd0P
2d0P3d0 =
h0Pe0P6e0 = h0P
2d30 = h0P
2e0P5e0 = h0P
3e0P4e0 =
h2d0P7e0 = h2e0P
7d0 = h2Pd0P6e0 = h2Pe0P
6d0 =h2P
2d0P5e0 = h2P
2e0P5d0 = h2P
3d0P4e0 = h2P
3e0P4d0 =
d0e0P7h2 = d0Ph2P
6e0 = d0Pe0P6h2 = d0P
2h2P5e0 =
d0P2e0P
5h2 = d0P3h2P
4e0 = d0P3e0P
4h2 = e0Ph2P6d0 =
e0Pd0P6h2 = e0P
2h2P5d0 = e0P
2d0P5h2 = e0P
3h2P4d0 =
e0P3d0P
4h2 = Ph2Pd0P5e0 = Ph2Pe0P
5d0 = Ph2P2d0P
4e0 =Ph2P
2e0P4d0 = Ph2P
3d0P3e0 = Pd0Pe0P
5h2 =Pd0P
2h2P4e0 = Pd0P
2e0P4h2 = Pd0P
3h2P3e0 =
Pe0P2h2P
4d0 = Pe0P2d0P
4h2 = Pe0P3h2P
3d0 =P 2h2P
2d0P3e0 = P 2h2P
2e0P3d0 = P 2d0P
2e0P3h2
38 (1) h20d
20P
6d0 = h20d0Pd0P
5d0 = h20d0P
2d0P4d0 = h2
0d0P3d2
0 =h2
0e0P7e0 = h2
0gP 7d0 = h20Pd2
0P4d0 = h2
0Pd0P2d0P
3d0 =h2
0Pe0P6e0 = h2
0P2d3
0 = h20P
2e0P5e0 = h2
0P3e0P
4e0 =h0h2d0P
7e0 = h0h2e0P7d0 = h0h2Pd0P
6e0 = h0h2Pe0P6d0 =
h0h2P2d0P
5e0 = h0h2P2e0P
5d0 = h0h2P3d0P
4e0 =h0h2P
3e0P4d0 = h0d0e0P
7h2 = h0d0Ph2P6e0 =
h0d0Pe0P6h2 = h0d0P
2h2P5e0 = h0d0P
2e0P5h2 =
h0d0P3h2P
4e0 = h0d0P3e0P
4h2 = h0e0Ph2P6d0 =
h0e0Pd0P6h2 = h0e0P
2h2P5d0 = h0e0P
2d0P5h2 =
h0e0P3h2P
4d0 = h0e0P3d0P
4h2 = h0Ph2Pd0P5e0 =
h0Ph2Pe0P5d0 = h0Ph2P
2d0P4e0 = h0Ph2P
2e0P4d0 =
h0Ph2P3d0P
3e0 = h0Pd0Pe0P5h2 = h0Pd0P
2h2P4e0 =
h0Pd0P2e0P
4h2 = h0Pd0P3h2P
3e0 = h0Pe0P2h2P
4d0 =h0Pe0P
2d0P4h2 = h0Pe0P
3h2P3d0 = h0P
2h2P2d0P
3e0 =h0P
2h2P2e0P
3d0 = h0P2d0P
2e0P3h2 = h2
2d0P7d0 =
h22Pd0P
6d0 = h22P
2d0P5d0 = h2
2P3d0P
4d0 = h2d20P
7h2 =h2d0Ph2P
6d0 = h2d0Pd0P6h2 = h2d0P
2h2P5d0 =
h2d0P2d0P
5h2 = h2d0P3h2P
4d0 = h2d0P3d0P
4h2 =h2gP 8h2 = h2Ph2Pd0P
5d0 = h2Ph2P2d0P
4d0 = h2Ph2P3d2
0 =h2Pd2
0P5h2 = h2Pd0P
2h2P4d0 = h2Pd0P
2d0P4h2 =
h2Pd0P3h2P
3d0 = h2P2h2P
2d0P3d0 = h2P
2d20P
3h2 = h4P9h2 =
c0P7j = d2
0Ph2P6h2 = d2
0P2h2P
5h2 = d20P
3h2P4h2 =
d0Ph22P
5d0 = d0Ph2Pd0P5h2 = d0Ph2P
2h2P4d0 =
d0Ph2P2d0P
4h2 = d0Ph2P3h2P
3d0 = d0Pd0P2h2P
4h2 =d0Pd0P
3h22 = d0P
2h22P
3d0 = d0P2h2P
2d0P3h2 = gPh2P
7h2 =gP 2h2P
6h2 = gP 3h2P5h2 = gP 4h2
2 = Ph22Pd0P
4d0 =Ph2
2P2d0P
3d0 = Ph2Pd20P
4h2 = Ph2Pd0P2h2P
3d0 =Ph2Pd0P
2d0P3h2 = Ph2P
2h2P2d2
0 = Pc0P6j = jP 7c0 =
Pd20P
2h2P3h2 = Pd0P
2h22P
2d0 = P 2c0P5j = PjP 6c0 =
P 3c0P4j = P 2jP 5c0 = P 4c0P
3j
39 (1) P 8j
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 103
Stem 916 (1) h2
4D3
8 (1) x8,75
9 (10) h6d20
(01) h0x8,75
10 (1) h0h6d20 = h2h6Pe0 = h6e0Ph2
11 (10) x11,61
(01) h20h6d
20 = h0h2h6Pe0 = h0h6e0Ph2 = h2
2h6Pd0 = h2h6d0Ph2
12 (01) h0x11,61
(11) rA′ = rA
13 (10) h1x12,55
(01) h20x11,61
14 (1) h30x11,61
15 (1) h40x11,61 = xR1 = rX1 = iG21
16 (1) h50x11,61 = h0xR1 = h0rX1 = h0iG21 = h3x15,42 = yx′
17 (100) g3n = gtm = nr2 = Nw
(001) h60x11,61 = h2
0xR1 = h20rX1 = h2
0iG21 = h0h3x15,42 = h0yx′ =h3rR1 = h3iX1 = h4x16,32 = g2Q
(011) d0e0B4 = gx13,35 = jB23 = kx10,27 = kx10,28 = lB21 = mQ1
18 (1) h70x11,61 = h3
0xR1 = h30rX1 = h3
0iG21 = h20h3x15,42 = h2
0yx′ =h0h3rR1 = h0h3iX1 = h0h4x16,32 = h0d0e0B4 = h0gx13,35 =h0g2Q = h0jB23 = h0kx10,27 = h0kx10,28 = h0lB21 = h0mQ1 =h2d
20B4 = h2e0x13,35 = h2gP 2D1 = h2iB23 = h2jx10,27 =
h2jx10,28 = h2kB21 = h2lQ1 = h2mx′ = h23x16,33 = d0f0B21 =
d0kB2 = e0f0Q1 = e0gR1 = e0jB2 = f0gx′ = gPh2B4 = giB2 =D2P
2e0 = AP 2d0 = Q2Pj = Pd0PA = Pe0B5 = Pe0PD2
continued
104 ROBERT R. BRUNER
Stem 91 continued19 (1) h8
0x11,61 = h40xR1 = h4
0rX1 = h40iG21 = h3
0h3x15,42 =h3
0yx′ = h20h3rR1 = h2
0h3iX1 = h20h4x16,32 = h2
0d0e0B4 =h2
0gx13,35 = h20g2Q = h2
0jB23 = h20kx10,27 = h2
0kx10,28 =h2
0lB21 = h20mQ1 = h0h2d
20B4 = h0h2e0x13,35 = h0h2gP 2D1 =
h0h2iB23 = h0h2jx10,27 = h0h2jx10,28 = h0h2kB21 = h0h2lQ1 =h0h2mx′ = h0h
23x16,33 = h0d0f0B21 = h0d0kB2 = h0e0f0Q1 =
h0e0gR1 = h0e0jB2 = h0f0gx′ = h0gPh2B4 = h0giB2 =h0D2P
2e0 = h0AP 2d0 = h0Q2Pj = h0Pd0PA = h0Pe0B5 =h0Pe0PD2 = h1d
20x10,27 = h1d
20x10,28 = h1d0e0B21 = h1d0gQ1 =
h1e20Q1 = h1e0gx′ = h1Pe0B23 = h2
2d0x13,35 = h22e0P
2D1 =h2
2gR2 = h22ix10,27 = h2
2ix10,28 = h22jB21 = h2
2kQ1 = h22lx
′ =h2
2Pe0B4 = h2h4PR2 = h2d0f0Q1 = h2d0gR1 = h2d0jB2 =h2e
20R1 = h2e0f0x
′ = h2e0Ph2B4 = h2e0iB2 = h2D2P2d0 =
h2Pd0B5 = h2Pd0PD2 = h23iR1 = h3xQ = h4Ph2R2 =
h4Pd0R1 = c0rx′ = c0iB4 = d2
0e0B1 = d0Ph2B5 = d0Ph2PD2 =d0D2P
2h2 = e0Ph1B23 = f0Ph2x10,27 = f0Ph2x10,28 =f0B2Pe0 = gPh1x10,27 = gPh1x10,28 = gB1Pe0 = Ph2D2Pd0 =Ph2iQ2 = Ph2lB2 = ryi = A′′P 3h2
20 (10) d0x16,33 = iP 2D1 = jR2
(01) d0e0g3 = d0e0r
2 = d0lw = d0mv = e30g
2 = e0kw = e0lv = e0mu =grz = gjw = gkv = glu = rjm = rkl
21 (1) h0d0x16,33 = h0iP2D1 = h0jR2 = h1d0x16,32 = h2iR2 = d0iR1 =
f0PR2 = Ph1x16,38 = Ph2x16,37 = Pd0W1
23 (1) d40m = d3
0e0l = d30gk = d2
0e20k = d2
0e0gj = d20g
2i = d0e30j =
d0e20gi = d0e0mPe0 = d0glPe0 = d0gmPd0 = d0rPu = e4
0i =e20lP e0 = e2
0mPd0 = e0g2Pj = e0gkPe0 = e0glPd0 = g2jPe0 =
g2kPd0 = rPd0u = i2w = ijv = iku = j2u = zPv
26 (10) Pd0x18,20
(01) d30ij = d2
0e0i2 = d2
0rP2e0 = d2
0kPj = d20Pd0z = d0e0rP
2d0 =d0e0jP j = d0giP j = d0rPd0Pe0 = d0ikPe0 = d0ilPd0 =d0j
2Pe0 = d0jkPd0 = d0mP 2j = e20iP j = e0rPd2
0 = e0ijPe0 =e0ikPd0 = e0j
2Pd0 = e0lP2j = e0mP 2i = e0zP 2e0 = grP 3e0 =
gi2Pe0 = gijPd0 = gkP 2j = glP 2i = gzP 2d0 = imP 2e0 =jlP 2e0 = jmP 2d0 = k2P 2e0 = klP 2d0 = lP e0Pj = mPd0Pj =Pe2
0z
(11) x′P 3d0
27 (1) h0Pd0x18,20 = h0x′P 3d0 = h2P
4Q1 = Ph2P3Q1 = P 2h2P
2Q1 =Q1 P 4h2 = P 3h2PQ1
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 105
Stem 91 continued28 (1) h2
0Pd0x18,20 = h20x
′P 3d0 = h0h2P4Q1 = h0Ph2P
3Q1 =h0P
2h2P2Q1 = h0Q1 P 4h2 = h0P
3h2PQ1 = h22P
4x′ =h2P
2h2x18,20 = h2x′P 4h2 = Ph2
2x18,20 = Ph2x′P 3h2 = B2P
5h2 =P 2h2
2x′
29 (1) d20P
3u = d0Pd0P2u = d0uP 3d0 = d0P
2d0Pu = e0P4v = gP 4u =
riP 3d0 = rPd0P2i = i3Pd0 = ijP 2j = ikP 2i = iP j2 = j2P 2i =
Pd20Pu = Pd0uP 2d0 = Pe0P
3v = vP 4e0 = wP 4d0 = zP 3j =P 2e0P
2v = PvP 3e0
32 (1) d30P
4e0 = d20e0P
4d0 = d20Pd0P
3e0 = d20Pe0P
3d0 = d20P
2d0P2e0 =
d0e0Pd0P3d0 = d0e0P
2d20 = d0gP 5e0 = d0Pd2
0P2e0 =
d0Pd0Pe0P2d0 = e2
0P5e0 = e0gP 5d0 = e0Pd2
0P2d0 =
e0Pe0P4e0 = e0P
2e0P3e0 = gPd0P
4e0 = gPe0P4d0 =
gP 2d0P3e0 = gP 2e0P
3d0 = Pd30Pe0 = Pe2
0P3e0 = Pe0P
2e20
35 (1) d0iP5d0 = d0Pd0P
4i = d0P2iP 3d0 = e0P
6j = gP 6i =iPd0P
4d0 = iP 2d0P3d0 = jP 6e0 = kP 6d0 = Pd0P
2d0P2i =
Pe0P5j = PjP 5e0 = P 2e0P
4j = P 2jP 4e0 = P 3e0P3j
36 (1) h0d0iP5d0 = h0d0Pd0P
4i = h0d0P2iP 3d0 = h0e0P
6j =h0gP 6i = h0iPd0P
4d0 = h0iP2d0P
3d0 = h0jP6e0 = h0kP 6d0 =
h0Pd0P2d0P
2i = h0Pe0P5j = h0PjP 5e0 = h0P
2e0P4j =
h0P2jP 4e0 = h0P
3e0P3j = h2d0P
6j = h2e0P6i = h2iP
6e0 =h2jP
6d0 = h2Pd0P5j = h2PjP 5d0 = h2P
2d0P4j = h2P
2e0P4i =
h2P2iP 4e0 = h2P
2jP 4d0 = h2P3d0P
3j = d0Ph2P5j =
d0jP6h2 = d0P
2h2P4j = d0PjP 5h2 = d0P
3h2P3j =
d0P2jP 4h2 = e0iP
6h2 = e0P2h2P
4i = e0P2iP 4h2 = f0P
7e0 =Ph2iP
5e0 = Ph2jP5d0 = Ph2Pd0P
4j = Ph2Pe0P4i =
Ph2PjP 4d0 = Ph2P2d0P
3j = Ph2P2iP 3e0 = Ph2P
2jP 3d0 =iPe0P
5h2 = iP 2h2P4e0 = iP 2e0P
4h2 = iP 3h2P3e0 =
jPd0P5h2 = jP 2h2P
4d0 = jP 2d0P4h2 = jP 3h2P
3d0 = lP 7h2 =Pd0P
2h2P3j = Pd0PjP 4h2 = Pd0P
3h2P2j = Pe0P
3h2P2i =
P 2h2PjP 3d0 = P 2h2P2d0P
2j = P 2h2P2e0P
2i = PjP 2d0P3h2
continued
106 ROBERT R. BRUNER
Stem 91 continued37 (1) h2
0d0iP5d0 = h2
0d0Pd0P4i = h2
0d0P2iP 3d0 = h2
0e0P6j =
h20gP 6i = h2
0iPd0P4d0 = h2
0iP2d0P
3d0 = h20jP
6e0 =h2
0kP 6d0 = h20Pd0P
2d0P2i = h2
0Pe0P5j = h2
0PjP 5e0 =h2
0P2e0P
4j = h20P
2jP 4e0 = h20P
3e0P3j = h0h2d0P
6j =h0h2e0P
6i = h0h2iP6e0 = h0h2jP
6d0 = h0h2Pd0P5j =
h0h2PjP 5d0 = h0h2P2d0P
4j = h0h2P2e0P
4i = h0h2P2iP 4e0 =
h0h2P2jP 4d0 = h0h2P
3d0P3j = h0d0Ph2P
5j = h0d0jP6h2 =
h0d0P2h2P
4j = h0d0PjP 5h2 = h0d0P3h2P
3j = h0d0P2jP 4h2 =
h0e0iP6h2 = h0e0P
2h2P4i = h0e0P
2iP 4h2 = h0f0P7e0 =
h0Ph2iP5e0 = h0Ph2jP
5d0 = h0Ph2Pd0P4j = h0Ph2Pe0P
4i =h0Ph2PjP 4d0 = h0Ph2P
2d0P3j = h0Ph2P
2iP 3e0 =h0Ph2P
2jP 3d0 = h0iPe0P5h2 = h0iP
2h2P4e0 = h0iP
2e0P4h2 =
h0iP3h2P
3e0 = h0jPd0P5h2 = h0jP
2h2P4d0 = h0jP
2d0P4h2 =
h0jP3h2P
3d0 = h0lP7h2 = h0Pd0P
2h2P3j = h0Pd0PjP 4h2 =
h0Pd0P3h2P
2j = h0Pe0P3h2P
2i = h0P2h2PjP 3d0 =
h0P2h2P
2d0P2j = h0P
2h2P2e0P
2i = h0PjP 2d0P3h2 =
h1d20P
6d0 = h1d0Pd0P5d0 = h1d0P
2d0P4d0 = h1d0P
3d20 =
h1e0P7e0 = h1gP 7d0 = h1Pd2
0P4d0 = h1Pd0P
2d0P3d0 =
h1Pe0P6e0 = h1P
2d30 = h1P
2e0P5e0 = h1P
3e0P4e0 =
h22d0P
6i = h22iP
6d0 = h22P
2d0P4i = h2
2P2iP 4d0 =
h2d0iP6h2 = h2d0P
2h2P4i = h2d0P
2iP 4h2 = h2f0P7d0 =
h2Ph2iP5d0 = h2Ph2Pd0P
4i = h2Ph2P2iP 3d0 = h2iPd0P
5h2 =h2iP
2h2P4d0 = h2iP
2d0P4h2 = h2iP
3h2P3d0 = h2kP 7h2 =
h2Pd0P3h2P
2i = h2P2h2P
2d0P2i = d3
0P6h1 = d2
0Ph1P5d0 =
d20Pd0P
5h1 = d20P
2h1P4d0 = d2
0P2d0P
4h1 = d20P
3h1P3d0 =
d0f0P7h2 = d0gP 7h1 = d0Ph1Pd0P
4d0 = d0Ph1P2d0P
3d0 =d0Ph2
2P4i = d0Ph2iP
5h2 = d0Ph2P3h2P
2i = d0iP2h2P
4h2 =d0iP
3h22 = d0Pd2
0P4h1 = d0Pd0P
2h1P3d0 = d0Pd0P
2d0P3h1 =
d0P2h1P
2d20 = d0P
2h22P
2i = e20P
7h1 = e0Ph1P6e0 =
e0Pe0P6h1 = e0P
2h1P5e0 = e0P
2e0P5h1 = e0P
3h1P4e0 =
e0P3e0P
4h1 = f0Ph2P6d0 = f0Pd0P
6h2 = f0P2h2P
5d0 =f0P
2d0P5h2 = f0P
3h2P4d0 = f0P
3d0P4h2 = gPh1P
6d0 =gPd0P
6h1 = gP 2h1P5d0 = gP 2d0P
5h1 = gP 3h1P4d0 =
gP 3d0P4h1 = Ph1Pd2
0P3d0 = Ph1Pd0P
2d20 = Ph1Pe0P
5e0 =Ph1P
2e0P4e0 = Ph1P
3e20 = Ph2
2iP4d0 = Ph2
2P2d0P
2i =Ph2iPd0P
4h2 = Ph2iP2h2P
3d0 = Ph2iP2d0P
3h2 =Ph2kP 6h2 = Ph2Pd0P
2h2P2i = iPd0P
2h2P3h2 = iP 2h2
2P2d0 =
kP 2h2P5h2 = kP 3h2P
4h2 = Pd30P
3h1 = Pd20P
2h1P2d0 =
Pe20P
5h1 = Pe0P2h1P
4e0 = Pe0P2e0P
4h1 = Pe0P3h1P
3e0 =P 2h1P
2e0P3e0 = P 2e2
0P3h1
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 107
Stem 924 (1) g3
5 (1) h0g3
6 (1) h20g3
7 (10) h1h24D3 = h3x6,68 = h5A
′ = c0f2
(01) h30g3
8 (10) h6k
(01) h40g3 = h4x7,57
9 (100) h1x8,75
(010) h0h6k = h2h6j
(001) h50g3 = h0h4x7,57
10 (10) x10,65
(01) h20h6k = h0h2h6j = h1h6d
20 = h2
2h6i = h6f0Ph2 = h6gPh1
11 (10) nA′ = nA = H1r
(01) h0x10,65
12 (10) x12,58
(01) h20x10,65 = h1x11,61
13 (1) h0x12,58
14 (100) rx8,33 = mQ2
(010) h21x12,55 = B2
1
(001) h20x12,58 = h3x13,46
16 (1) rx10,27 = rx10,28 = lB4
18 (1) x18,55
19 (10) e0g2m = e0rw = g3l = grv = r2l
(01) ux′
22 (01) d30gr = d2
0e20r = d2
0km = d20l
2 = d0e0gz = d0e0jm = d0e0kl =d0gim = d0gjl = d0gk2 = d0u
2 = e30z = e2
0im = e20jl = e2
0k2 =
e0grPe0 = e0gil = e0gjk = g2rPd0 = g2ik = g2j2 = riu =lmPe0 = m2Pd0 = vPv = wPu
(11) d20PQ1 = d0Pd0Q1 = d0Pe0x
′ = e0Pd0x′ = gP 2Q1 =
B21P2e0 = x10,27P
2d0 = x10,28P2d0
23 (1) h0d20PQ1 = h0d0Pd0Q1 = h0d0Pe0x
′ = h0e0Pd0x′ =
h0gP 2Q1 = h0B21P2e0 = h0x10,27P
2d0 = h0x10,28P2d0 =
h2d0Pd0x′ = h2e0P
2Q1 = h2gx18,20 = h2Pe0PQ1 =h2Q1 P 2e0 = h2B21P
2d0 = d20Ph2x
′ = d0B2P2d0 = d0P
2h2B21 =e0Ph2PQ1 = e0P
2h2Q1 = gP 2h2x′ = Ph2Pd0B21 =
Ph2Pe0Q1 = Pc0x16,32 = B2Pd20 = B23P
3h2
24 (1) P 2x16,32
continued
108 ROBERT R. BRUNER
Stem 92 continued25 (100) x′P 2i
(001) h0P2x16,32
(101) Pd0R1
(011) d30Pv = d2
0e0Pu = d20rPj = d2
0iz = d20Pd0v = d2
0Pe0u =d0e0Pd0u = d0gP 2v = d0riPe0 = d0rjPd0 = d0i
2l = d0ijk =d0j
3 = d0wP 2e0 = e20P
2v = e0gP 2u = e0riPd0 = e0i2k = e0ij
2 =e0Pe0Pv = e0vP 2e0 = e0wP 2d0 = e0zPj = grP 2j = gi2j =gPd0Pv = gPe0Pu = guP 2e0 = gvP 2d0 = rkP 2e0 = rlP 2d0 =imPj = jlP j = jPe0z = k2Pj = kPd0z = Pd0Pe0w = Pe2
0v
(111) ix18,20
26 (10) h0ix18,20 = h0x′P 2i
(01) h20P
2x16,32
(11) h0Pd0R1 = h2P3R2 = Ph2P
2R2 = P 2h2PR2 = R1P3d0 =
P 3h2R2
27 (10) h20Pd0R1 = h0h2P
3R2 = h0Ph2P2R2 = h0P
2h2PR2 =h0R1P
3d0 = h0P3h2R2 = h1Pd0x18,20 = h1x
′P 3d0 = h22x25,24 =
h2P2h2R1 = d0Ph1x18,20 = d0x
′P 3h1 = Ph1x′P 2d0 = Ph2
2R1 =B1P
4d0 = Pd0P2h1x
′ = B21P4h1
(01) h30P
2x16,32 = h3P4x′
(11) h20ix18,20 = h2
0x′P 2i
28 (10) d50Pd0 = d3
0e0P2e0 = d3
0gP 2d0 = d30Pe2
0 = d20e
20P
2d0 =d20e0Pd0Pe0 = d2
0gPd20 = d0e
20Pd2
0 = d0e0gP 3e0 = d0g2P 3d0 =
d0gPe0P2e0 = d0iP
2u = d0uP 2i = e30P
3e0 = e20gP 3d0 =
e20Pe0P
2e0 = e0gPd0P2e0 = e0gPe0P
2d0 = e0Pe30 =
g2Pd0P2d0 = gPd0Pe2
0 = iPd0Pu = iuP 2d0 = jP 3v = kP 3u =vP 3j = PjP 2v = PvP 2j
(01) h40P
2x16,32 = h30ix18,20 = h3
0x′P 2i = h0h3P
4x′ = xP 4i
(11) riP 2i = i4
29 (1) h50P
2x16,32 = h40ix18,20 = h4
0x′P 2i = h2
0h3P4x′ = h0xP 4i =
h0riP2i = h0i
4
30 (1) h60P
2x16,32 = h50ix18,20 = h5
0x′P 2i = h3
0h3P4x′ = h2
0xP 4i =h2
0riP2i = h2
0i4 = h3rP
4i = h3i2P 2i
31 (1) d30P
3j = d20iP
3e0 = d20jP
3d0 = d20Pd0P
2j = d20Pe0P
2i =d20PjP 2d0 = d0e0iP
3d0 = d0e0Pd0P2i = d0gP 4j = d0iPd0P
2e0 =d0iPe0P
2d0 = d0jPd0P2d0 = d0kP 4e0 = d0lP
4d0 = d0Pd20Pj =
e20P
4j = e0gP 4i = e0iPd0P2d0 = e0jP
4e0 = e0kP 4d0 =e0Pe0P
3j = e0PjP 3e0 = e0P2e0P
2j = giP 4e0 = gjP 4d0 =gPd0P
3j = gPjP 3d0 = gP 2d0P2j = gP 2e0P
2i = iPd20Pe0 =
jPd30 = jPe0P
3e0 = jP 2e20 = kPd0P
3e0 = kPe0P3d0 =
kP 2d0P2e0 = lPd0P
3d0 = lP 2d20 = mP 5e0 = Pe2
0P2j =
Pe0PjP 2e0
34 (1) d0iP4i = d0P
2i2 = rP 6d0 = i2P 4d0 = iP 2d0P2i = jP 5j =
PjP 4j = P 2jP 3j
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 109
Stem 933 (1) h2
4h6 = h35
4 (1) h0h24h6 = h0h
35
5 (01) h20h
24h6 = h2
0h35
(11) h1g3
6 (1) h30h
24h6 = h3
0h35
7 (1) h6r
8 (10) x8,78
(01) h0h6r
9 (1) h20h6r = h0x8,78 = h3h6i
10 (100) x10,67
(010) h21x8,75 = h4x9,55 = d1A
′ = d1A = D3q = nH1
(001) h30h6r = h2
0x8,78 = h0h3h6i
11 (10) h1x10,65
(01) h40h6r = h3
0x8,78 = h20h3h6i = h0x10,67 = h4P
2h25
12 (10) x12,60
(01) h50h6r = h4
0x8,78 = h30h3h6i = h2
0x10,67 = h0h4P2h2
5 = h6c0Pd0 =h6d0Pc0
13 (0010) h1x12,58 = h3x12,48
(0110) e0x9,51 = nx8,33 = rx7,34 = tQ2 = D2m = A′l = Al
(0001) h0x12,60 = h2x12,55
(1001) h6P2d0
14 (01) h20x12,60 = h0h2x12,55
(11) h0h6P2d0
15 (10) h3P3h2
5
(01) h30x12,60 = h2
0h2x12,55 = h31x12,55 = h1B
21 = Ph1Px6,53 = xQ1 =
qX1 = Gu = x6,53P2h1
(11) h20h6P
2d0 = h2h6P3h2 = h6Ph2P
2h2
17 (1) x17,57
18 (1) d20B23 = d0e0x10,27 = d0e0x10,28 = d0gB21 = e2
0B21 = e0gQ1 =g2x′
20 (10) d0x16,35
(01) h1ux′ = Ph1x15,42 = Ph1x15,43 = B1Q = B1Pu
(11) e0x16,32
21 (10) d20R2 = d0iQ1 = d0jx
′ = e0ix′ = gPR2 = kPQ1 = Pd0x13,35 =
Pe0P2D1 = B4P
2e0 = B21Pj
(01) d20gw = d2
0rm = d0e20w = d0e0gv = d0e0rl = d0g
2u = d0grk =e30v = e2
0gu = e20rk = e0grj = e0mz = g2ri = glz = im2 = jlm =
k2m = kl2
continued
110 ROBERT R. BRUNER
Stem 93 continued22 (1) h0d
20R2 = h0d0iQ1 = h0d0jx
′ = h0e0ix′ = h0gPR2 =
h0kPQ1 = h0Pd0x13,35 = h0Pe0P2D1 = h0B4P
2e0 =h0B21Pj = h2d0ix
′ = h2e0PR2 = h2gR1 = h2jPQ1 =h2Pd0P
2D1 = h2Pe0R2 = h2B4P2d0 = h2Q1 Pj =
d0Ph2P2D1 = d0Pe0R1 = d0P
2h2B4 = e0Ph2R2 = e0Pd0R1 =f0Pd0x
′ = Ph2iB21 = Ph2jQ1 = Ph2kx′ = Ph2Pd0B4 =iB2Pd0
23 (1) h20d
20R2 = h2
0d0iQ1 = h20d0jx
′ = h20e0ix
′ = h20gPR2 =
h20kPQ1 = h2
0Pd0x13,35 = h20Pe0P
2D1 = h20B4P
2e0 =h2
0B21Pj = h0h2d0ix′ = h0h2e0PR2 = h0h2gR1 = h0h2jPQ1 =
h0h2Pd0P2D1 = h0h2Pe0R2 = h0h2B4P
2d0 = h0h2Q1 Pj =h0d0Ph2P
2D1 = h0d0Pe0R1 = h0d0P2h2B4 = h0e0Ph2R2 =
h0e0Pd0R1 = h0f0Pd0x′ = h0Ph2iB21 = h0Ph2jQ1 =
h0Ph2kx′ = h0Ph2Pd0B4 = h0iB2Pd0 = h1d20PQ1 =
h1d0Pd0Q1 = h1d0Pe0x′ = h1e0Pd0x
′ = h1gP 2Q1 =h1B21P
2e0 = h1x10,27P2d0 = h1x10,28P
2d0 = h22d0PR2 =
h22e0R1 = h2
2iPQ1 = h22Pd0R2 = h2
2x′Pj = h2d0Ph2R2 =
h2d0Pd0R1 = h2f0P2Q1 = h2Ph2iQ1 = h2Ph2jx
′ =h2B2P
2j = h2P2h2x13,35 = d2
0Ph1Q1 = d20Ph2R1 = d0e0Ph1x
′ =d0B1P
2e0 = d0P2h1x10,27 = d0P
2h1x10,28 = e0B1P2d0 =
e0P2h1B21 = f0Ph2PQ1 = f0P
2h2Q1 = gPh1PQ1 =gP 2h1Q1 = gP 2h2R1 = Ph1Pd0x10,27 = Ph1Pd0x10,28 =Ph1Pe0B21 = Ph2
2x13,35 = Ph2B2Pj = D2P4h2 = Pc0x16,33 =
jB2P2h2 = B1Pd0Pe0 = B5 P 3h2 = PD2P
3h2
24 (10) iR1
(01) d40e0g = d3
0e30 = d2
0g2Pe0 = d2
0iv = d20ju = d0e
20gPe0 =
d0e0g2Pd0 = d0e0iu = d0rij = d0kPv = d0lPu = d0wPj =
e40Pe0 = e3
0gPd0 = e0ri2 = e0jPv = e0kPu = e0vPj = g3P 2e0 =
giPv = gjPu = guPj = r2P 2e0 = rkPj = rPd0z = ikz =iPe0w = j2z = jPd0w = jPe0v = kPd0v = kPe0u = lPd0u =mP 2v
25 (01) h0iR1 = R1P2i
(11) h1P2x16,32 = P 2h1x16,32
26 (1) h20iR1 = h0R1P
2i = h3x25,24
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 111
Stem 93 continued27 (10) d5
0i = d30e0Pj = d3
0jPe0 = d30kPd0 = d2
0e0iPe0 = d20e0jPd0 =
d20giPd0 = d2
0lP2e0 = d2
0mP 2d0 = d0e20iPd0 = d0e0gP 2j =
d0e0kP 2e0 = d0e0lP2d0 = d0g
2P 2i = d0gjP 2e0 = d0gkP 2d0 =d0gPe0Pj = d0kPe2
0 = d0lPd0Pe0 = d0mPd20 = e3
0P2j =
e20gP 2i = e2
0jP2e0 = e2
0kP 2d0 = e20Pe0Pj = e0giP 2e0 =
e0gjP 2d0 = e0gPd0Pj = e0jPe20 = e0kPd0Pe0 = e0lPd2
0 =e0mP 3e0 = g2iP 2d0 = giPe2
0 = gjPd0Pe0 = gkPd20 = glP 3e0 =
gmP 3d0 = rP 3u = i2Pu = mPe0P2e0
(01) h30iR1 = h2
0R1P2i = h0h3x25,24 = i2Q
28 (1) h40iR1 = h3
0R1P2i = h2
0h3x25,24 = h0i2Q
29 (1) h50iR1 = h4
0R1P2i = h3
0h3x25,24 = h20i
2Q = h3QP 2i = c0P4x′ =
yP 4i = x′P 4c0 = P 2c0x18,20
30 (1) d20iP
2j = d20jP
2i = d0e0iP2i = d0rP
4e0 = d0i2P 2e0 =
d0ijP2d0 = d0iPd0Pj = d0kP 3j = d0zP 3d0 = e0rP
4d0 =e0i
2P 2d0 = e0jP3j = e0PjP 2j = giP 3j = gPjP 2i = rPd0P
3e0 =rPe0P
3d0 = rP 2d0P2e0 = i2Pd0Pe0 = ijPd2
0 = ikP 3e0 =ilP 3d0 = j2P 3e0 = jkP 3d0 = jPe0P
2j = jP jP 2e0 = kPd0P2j =
kPe0P2i = kPjP 2d0 = lPd0P
2i = mP 4j = Pd0zP 2d0 = Pe0Pj2
33 (1) d0P5u = Pd0P
4u = uP 5d0 = P 2d0P3u = PuP 4d0 = P 3d0P
2u
36 (1) d20P
6e0 = d0e0P6d0 = d0Pd0P
5e0 = d0Pe0P5d0 = d0P
2d0P4e0 =
d0P2e0P
4d0 = d0P3d0P
3e0 = e0Pd0P5d0 = e0P
2d0P4d0 =
e0P3d2
0 = gP 7e0 = Pd20P
4e0 = Pd0Pe0P4d0 = Pd0P
2d0P3e0 =
Pd0P2e0P
3d0 = Pe0P2d0P
3d0 = P 2d20P
2e0
39 (1) iP 7d0 = Pd0P6i = P 2iP 5d0 = P 3d0P
4i
112 ROBERT R. BRUNER
Stem 944 (1) h1h
24h6 = h1h
35
6 (10) h6n
(01) h21g3 = h4x1
8 (1) x8,80
9 (001) h0x8,80
(101) h6d0e0
(011) h1x8,78 = h2x8,75 = d1H1
10 (100) x10,70
(001) h20x8,80 = pA′
(011) h0h6d0e0 = h2h6d20 = h6gPh2
11 (10) h1x10,67 = h3x10,60 = H1q
(01) h20h6d0e0 = h0h2h6d
20 = h0h6gPh2 = h2
2h6Pe0 = h2h6e0Ph2 =h6c0i
12 (01) h21x10,65 = d1x8,32 = xQ2
(11) h2x11,61
13 (1) h0h2x11,61 = h1x12,60
14 (1) h1h6P2d0 = h6d0P
2h1 = h6Ph1Pd0
15 (1) x15,56
16 (1) x16,54
17 (1) d0gB4 = e20B4 = kB23 = lx10,27 = lx10,28 = mB21
19 (1) x19,49
20 (10) d0g4 = d0gr2 = d0mw = e2
0g3 = e2
0r2 = e0lw = e0mv = gkw =
glv = gmu = rkm = rl2
(01) d0x16,37 = e0x16,33 = rPQ1 = ix13,35 = jP 2D1 = kR2 = B4Pj =zx′
21 (1) h0d0x16,37 = h0e0x16,33 = h0rPQ1 = h0ix13,35 = h0jP2D1 =
h0kR2 = h0B4Pj = h0zx′ = h21ux′ = h1d0x16,35 = h1Ph1x15,42 =
h1Ph1x15,43 = h1B1Q = h1B1Pu = h2d0x16,33 = h2iP2D1 =
h2jR2 = h3Px16,35 = d0jR1 = e0iR1 = f0ix′ = D3P
4h1 =Ph1qx
′ = Ph1B1u = Ph1x16,42 = Ph2rx′ = Ph2iB4 = i2B2 =
PD3P3h1 = P 2h1P
2D3 = X1P2e0
22 (1) x22,39
23 (10) d30e0m = d3
0gl = d20e
20l = d2
0e0gk = d20g
2j = d0e30k = d0e
20gj =
d0e0g2i = d0gmPe0 = d0rPv = d0uz = e4
0j = e30gi = e2
0mPe0 =e0glPe0 = e0gmPd0 = e0rPu = g3Pj = g2kPe0 = g2lPd0 =r2Pj = riz = rPd0v = rPe0u = ijw = ikv = ilu = j2v = jku
(01) h0x22,39
24 (1) h20x22,39
25 (1) h30x22,39
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 113
Stem 94 continued26 (0011) Q2
(1011) d0P3Q1 = Pd0P
2Q1 = Q1 P 3d0 = P 2d0PQ1
(0100) uP 2u = Pu2
(0010) h21P
2x16,32 = h1P2h1x16,32 = Ph2
1x16,32 = X1P4h1 = P 3h1W1 =
QPu
(1111) Pe0x18,20
(0110) d30rPd0 = d3
0ik = d30j
2 = d20e0ij = d2
0gi2 = d20lP j = d2
0Pe0z =d0e
20i
2 = d0e0rP2e0 = d0e0kPj = d0e0Pd0z = d0grP 2d0 =
d0gjPj = d0rPe20 = d0ilPe0 = d0imPd0 = d0jkPe0 = d0jlPd0 =
d0k2Pd0 = e2
0rP2d0 = e2
0jP j = e0giP j = e0rPd0Pe0 =e0ikPe0 = e0ilPd0 = e0j
2Pe0 = e0jkPd0 = e0mP 2j = grPd20 =
gijPe0 = gikPd0 = gj2Pd0 = glP 2j = gmP 2i = gzP 2e0 =jmP 2e0 = klP 2e0 = kmP 2d0 = l2P 2d0 = mPe0Pj
(0001) h40x22,39
(1001) x′P 3e0
27 (01) h50x22,39 = h0Q
2
(11) h0d0P3Q1 = h0Pd0P
2Q1 = h0Pe0x18,20 = h0x′P 3e0 =
h0Q1 P 3d0 = h0P2d0PQ1 = h2Pd0x18,20 = h2x
′P 3d0 =d0Ph2x18,20 = d0x
′P 3h2 = Ph2x′P 2d0 = B2P
4d0 = Pd0P2h2x
′ =B21P
4h2
28 (10) h20d0P
3Q1 = h20Pd0P
2Q1 = h20Pe0x18,20 = h2
0x′P 3e0 =
h20Q1 P 3d0 = h2
0P2d0PQ1 = h0h2Pd0x18,20 = h0h2x
′P 3d0 =h0d0Ph2x18,20 = h0d0x
′P 3h2 = h0Ph2x′P 2d0 = h0B2P
4d0 =h0Pd0P
2h2x′ = h0B21P
4h2 = h22P
4Q1 = h2Ph2P3Q1 =
h2P2h2P
2Q1 = h2Q1 P 4h2 = h2P3h2PQ1 = c0x25,24 =
Ph22P
2Q1 = Ph2P2h2PQ1 = Ph2Q1 P 3h2 = P 2h2
2Q1 =P 2c0R1
(01) h60x22,39 = h2
0Q2
29 (10) d20P
3v = d0e0P3u = d0rP
3j = d0i2Pj = d0Pd0P
2v =d0Pe0P
2u = d0uP 3e0 = d0vP 3d0 = d0zP 2i = d0P2d0Pv =
d0P2e0Pu = e0Pd0P
2u = e0uP 3d0 = e0P2d0Pu = gP 4v =
riP 3e0 = rjP 3d0 = rPd0P2j = rPe0P
2i = rPjP 2d0 = i3Pe0 =i2jPd0 = ikP 2j = ilP 2i = izP 2d0 = j2P 2j = jkP 2i = jP j2 =Pd2
0Pv = Pd0Pe0Pu = Pd0uP 2e0 = Pd0vP 2d0 = Pe0uP 2d0 =wP 4e0
(01) h70x22,39 = h3
0Q2
30 (1) h80x22,39 = h4
0Q2 = h4P
4Q
31 (10) Ph1P4x′ = x′P 5h1 = P 3h1x18,20
(01) h90x22,39 = h5
0Q2 = h0h4P
4Q
continued
114 ROBERT R. BRUNER
Stem 94 continued32 (10) d4
0P3d0 = d3
0Pd0P2d0 = d2
0e0P4e0 = d2
0gP 4d0 = d20Pd3
0 =d20Pe0P
3e0 = d20P
2e20 = d0e
20P
4d0 = d0e0Pd0P3e0 =
d0e0Pe0P3d0 = d0e0P
2d0P2e0 = d0gPd0P
3d0 = d0gP 2d20 =
d0Pd0Pe0P2e0 = d0Pe2
0P2d0 = e2
0Pd0P3d0 = e2
0P2d2
0 =e0gP 5e0 = e0Pd2
0P2e0 = e0Pd0Pe0P
2d0 = g2P 5d0 =gPd2
0P2d0 = gPe0P
4e0 = gP 2e0P3e0 = iP 4u = Pd2
0Pe20 =
uP 4i = P 2iP 2u
(01) h100 x22,39 = h6
0Q2 = h2
0h4P4Q
33 (1) h110 x22,39 = h7
0Q2 = h3
0h4P4Q
34 (1) h120 x22,39 = h8
0Q2 = h4
0h4P4Q
35 (01) h130 x22,39 = h9
0Q2 = h5
0h4P4Q
(11) d20P
5j = d0iP5e0 = d0jP
5d0 = d0Pd0P4j = d0Pe0P
4i =d0PjP 4d0 = d0P
2d0P3j = d0P
2iP 3e0 = d0P2jP 3d0 = e0iP
5d0 =e0Pd0P
4i = e0P2iP 3d0 = gP 6j = iPd0P
4e0 = iPe0P4d0 =
iP 2d0P3e0 = iP 2e0P
3d0 = jPd0P4d0 = jP 2d0P
3d0 = kP 6e0 =lP 6d0 = Pd2
0P3j = Pd0PjP 3d0 = Pd0P
2d0P2j = Pd0P
2e0P2i =
Pe0P2d0P
2i = PjP 2d20
36 (1) h140 x22,39 = h10
0 Q2 = h60h4P
4Q = h0d20P
5j = h0d0iP5e0 =
h0d0jP5d0 = h0d0Pd0P
4j = h0d0Pe0P4i = h0d0PjP 4d0 =
h0d0P2d0P
3j = h0d0P2iP 3e0 = h0d0P
2jP 3d0 = h0e0iP5d0 =
h0e0Pd0P4i = h0e0P
2iP 3d0 = h0gP 6j = h0iPd0P4e0 =
h0iPe0P4d0 = h0iP
2d0P3e0 = h0iP
2e0P3d0 = h0jPd0P
4d0 =h0jP
2d0P3d0 = h0kP 6e0 = h0lP
6d0 = h0Pd20P
3j =h0Pd0PjP 3d0 = h0Pd0P
2d0P2j = h0Pd0P
2e0P2i =
h0Pe0P2d0P
2i = h0PjP 2d20 = h2d0iP
5d0 = h2d0Pd0P4i =
h2d0P2iP 3d0 = h2e0P
6j = h2gP 6i = h2iPd0P4d0 =
h2iP2d0P
3d0 = h2jP6e0 = h2kP 6d0 = h2Pd0P
2d0P2i =
h2Pe0P5j = h2PjP 5e0 = h2P
2e0P4j = h2P
2jP 4e0 =h2P
3e0P3j = d2
0Ph2P4i = d2
0iP5h2 = d2
0P3h2P
2i = d0f0P6d0 =
d0Ph2iP4d0 = d0Ph2P
2d0P2i = d0iPd0P
4h2 = d0iP2h2P
3d0 =d0iP
2d0P3h2 = d0kP 6h2 = d0Pd0P
2h2P2i = e0Ph2P
5j =e0jP
6h2 = e0P2h2P
4j = e0PjP 5h2 = e0P3h2P
3j =e0P
2jP 4h2 = f0Pd0P5d0 = f0P
2d0P4d0 = f0P
3d20 = giP 6h2 =
gP 2h2P4i = gP 2iP 4h2 = Ph2iPd0P
3d0 = Ph2iP2d2
0 =Ph2jP
5e0 = Ph2kP 5d0 = Ph2Pd20P
2i = Ph2Pe0P4j =
Ph2PjP 4e0 = Ph2P2e0P
3j = Ph2P2jP 3e0 = iPd2
0P3h2 =
iPd0P2h2P
2d0 = jPe0P5h2 = jP 2h2P
4e0 = jP 2e0P4h2 =
jP 3h2P3e0 = kPd0P
5h2 = kP 2h2P4d0 = kP 2d0P
4h2 =kP 3h2P
3d0 = mP 7h2 = Pe0P2h2P
3j = Pe0PjP 4h2 =Pe0P
3h2P2j = P 2h2PjP 3e0 = P 2h2P
2e0P2j = PjP 2e0P
3h2
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 115
Stem 94 continued37 (1) h15
0 x22,39 = h110 Q2 = h7
0h4P4Q = h2
0d20P
5j = h20d0iP
5e0 =h2
0d0jP5d0 = h2
0d0Pd0P4j = h2
0d0Pe0P4i = h2
0d0PjP 4d0 =h2
0d0P2d0P
3j = h20d0P
2iP 3e0 = h20d0P
2jP 3d0 = h20e0iP
5d0 =h2
0e0Pd0P4i = h2
0e0P2iP 3d0 = h2
0gP 6j = h20iPd0P
4e0 =h2
0iPe0P4d0 = h2
0iP2d0P
3e0 = h20iP
2e0P3d0 = h2
0jPd0P4d0 =
h20jP
2d0P3d0 = h2
0kP 6e0 = h20lP
6d0 = h20Pd2
0P3j =
h20Pd0PjP 3d0 = h2
0Pd0P2d0P
2j = h20Pd0P
2e0P2i =
h20Pe0P
2d0P2i = h2
0PjP 2d20 = h0h2d0iP
5d0 = h0h2d0Pd0P4i =
h0h2d0P2iP 3d0 = h0h2e0P
6j = h0h2gP 6i = h0h2iPd0P4d0 =
h0h2iP2d0P
3d0 = h0h2jP6e0 = h0h2kP 6d0 = h0h2Pd0P
2d0P2i =
h0h2Pe0P5j = h0h2PjP 5e0 = h0h2P
2e0P4j = h0h2P
2jP 4e0 =h0h2P
3e0P3j = h0d
20Ph2P
4i = h0d20iP
5h2 = h0d20P
3h2P2i =
h0d0f0P6d0 = h0d0Ph2iP
4d0 = h0d0Ph2P2d0P
2i =h0d0iPd0P
4h2 = h0d0iP2h2P
3d0 = h0d0iP2d0P
3h2 =h0d0kP 6h2 = h0d0Pd0P
2h2P2i = h0e0Ph2P
5j = h0e0jP6h2 =
h0e0P2h2P
4j = h0e0PjP 5h2 = h0e0P3h2P
3j = h0e0P2jP 4h2 =
h0f0Pd0P5d0 = h0f0P
2d0P4d0 = h0f0P
3d20 = h0giP 6h2 =
h0gP 2h2P4i = h0gP 2iP 4h2 = h0Ph2iPd0P
3d0 = h0Ph2iP2d2
0 =h0Ph2jP
5e0 = h0Ph2kP 5d0 = h0Ph2Pd20P
2i = h0Ph2Pe0P4j =
h0Ph2PjP 4e0 = h0Ph2P2e0P
3j = h0Ph2P2jP 3e0 =
h0iPd20P
3h2 = h0iPd0P2h2P
2d0 = h0jPe0P5h2 =
h0jP2h2P
4e0 = h0jP2e0P
4h2 = h0jP3h2P
3e0 = h0kPd0P5h2 =
h0kP 2h2P4d0 = h0kP 2d0P
4h2 = h0kP 3h2P3d0 = h0mP 7h2 =
h0Pe0P2h2P
3j = h0Pe0PjP 4h2 = h0Pe0P3h2P
2j =h0P
2h2PjP 3e0 = h0P2h2P
2e0P2j = h0PjP 2e0P
3h2 =h1d
20P
6e0 = h1d0e0P6d0 = h1d0Pd0P
5e0 = h1d0Pe0P5d0 =
h1d0P2d0P
4e0 = h1d0P2e0P
4d0 = h1d0P3d0P
3e0 =h1e0Pd0P
5d0 = h1e0P2d0P
4d0 = h1e0P3d2
0 = h1gP 7e0 =h1Pd2
0P4e0 = h1Pd0Pe0P
4d0 = h1Pd0P2d0P
3e0 =h1Pd0P
2e0P3d0 = h1Pe0P
2d0P3d0 = h1P
2d20P
2e0 =h2
2d0P6j = h2
2e0P6i = h2
2iP6e0 = h2
2jP6d0 = h2
2Pd0P5j =
h22PjP 5d0 = h2
2P2d0P
4j = h22P
2e0P4i = h2
2P2iP 4e0 =
h22P
2jP 4d0 = h22P
3d0P3j = h2d0Ph2P
5j = h2d0jP6h2 =
h2d0P2h2P
4j = h2d0PjP 5h2 = h2d0P3h2P
3j = h2d0P2jP 4h2 =
h2e0iP6h2 = h2e0P
2h2P4i = h2e0P
2iP 4h2 = h2f0P7e0 =
h2Ph2iP5e0 = h2Ph2jP
5d0 = h2Ph2Pd0P4j = h2Ph2Pe0P
4i =h2Ph2PjP 4d0 = h2Ph2P
2d0P3j = h2Ph2P
2iP 3e0 =h2Ph2P
2jP 3d0 = h2iPe0P5h2 = h2iP
2h2P4e0 = h2iP
2e0P4h2 =
h2iP3h2P
3e0 = h2jPd0P5h2 = h2jP
2h2P4d0 = h2jP
2d0P4h2 =
h2jP3h2P
3d0 = h2lP7h2 = h2Pd0P
2h2P3j = h2Pd0PjP 4h2 =
h2Pd0P3h2P
2j = h2Pe0P3h2P
2i = (continued)
116 ROBERT R. BRUNER
Stem 86 continued37 (1) (continued) = h2P
2h2PjP 3d0 = h2P2h2P
2d0P2j =
h2P2h2P
2e0P2i = h2PjP 2d0P
3h2 = d20e0P
6h1 = d20Ph1P
5e0 =d20Pe0P
5h1 = d20P
2h1P4e0 = d2
0P2e0P
4h1 = d20P
3h1P3e0 =
d0e0Ph1P5d0 = d0e0Pd0P
5h1 = d0e0P2h1P
4d0 =d0e0P
2d0P4h1 = d0e0P
3h1P3d0 = d0Ph1Pd0P
4e0 =d0Ph1Pe0P
4d0 = d0Ph1P2d0P
3e0 = d0Ph1P2e0P
3d0 =d0Ph2
2P4j = d0Ph2jP
5h2 = d0Ph2P2h2P
3j = d0Ph2PjP 4h2 =d0Ph2P
3h2P2j = d0jP
2h2P4h2 = d0jP
3h22 = d0Pd0Pe0P
4h1 =d0Pd0P
2h1P3e0 = d0Pd0P
2e0P3h1 = d0Pe0P
2h1P3d0 =
d0Pe0P2d0P
3h1 = d0P2h1P
2d0P2e0 = d0P
2h22P
2j =d0P
2h2PjP 3h2 = e0f0P7h2 = e0gP 7h1 = e0Ph1Pd0P
4d0 =e0Ph1P
2d0P3d0 = e0Ph2
2P4i = e0Ph2iP
5h2 = e0Ph2P3h2P
2i =e0iP
2h2P4h2 = e0iP
3h22 = e0Pd2
0P4h1 = e0Pd0P
2h1P3d0 =
e0Pd0P2d0P
3h1 = e0P2h1P
2d20 = e0P
2h22P
2i = f0Ph2P6e0 =
f0Pe0P6h2 = f0P
2h2P5e0 = f0P
2e0P5h2 = f0P
3h2P4e0 =
f0P3e0P
4h2 = gPh1P6e0 = gPe0P
6h1 = gP 2h1P5e0 =
gP 2e0P5h1 = gP 3h1P
4e0 = gP 3e0P4h1 = Ph1Pd2
0P3e0 =
Ph1Pd0Pe0P3d0 = Ph1Pd0P
2d0P2e0 = Ph1Pe0P
2d20 =
Ph22iP
4e0 = Ph22jP
4d0 = Ph22Pd0P
3j = Ph22PjP 3d0 =
Ph22P
2d0P2j = Ph2
2P2e0P
2i = Ph2iPe0P4h2 =
Ph2iP2h2P
3e0 = Ph2iP2e0P
3h2 = Ph2jPd0P4h2 =
Ph2jP2h2P
3d0 = Ph2jP2d0P
3h2 = Ph2lP6h2 =
Ph2Pd0P2h2P
2j = Ph2Pd0PjP 3h2 = Ph2Pe0P2h2P
2i =Ph2P
2h2PjP 2d0 = rP 7c0 = Pc0iP4i = Pc0P
2i2 = i2P 5c0 =iPe0P
2h2P3h2 = iP 2h2
2P2e0 = iP 3c0P
2i = jPd0P2h2P
3h2 =jP 2h2
2P2d0 = lP 2h2P
5h2 = lP 3h2P4h2 = Pd2
0Pe0P3h1 =
Pd20P
2h1P2e0 = Pd0Pe0P
2h1P2d0 = Pd0P
2h22Pj
38 (1) iP 6i = P 2iP 4i
39 (1) h0iP6i = h0P
2iP 4i
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 117
Stem 955 (10) h6d1
(01) h2g3 = h4e2
6 (1) h0h2g3 = h0h4e2
7 (10) x7,79
(01) h6q
8 (10) h6l
(01) h0x7,79
9 (10) h1x8,80
(01) h0h6l = h2h6k = h6d0f0
10 (10) x10,73
(01) h20h6l = h0h2h6k = h0h6d0f0 = h1h6d0e0 = h2
2h6j
11 (10) h2x10,65
(01) h0x10,73 = e1Q2 = xD2
13 (1) h2x12,58 = h3x12,51 = yQ2
14 (1) h0h2x12,58 = h0h3x12,51 = h0yQ2 = h23x12,45 = px10,27
15 (1) h21h6P
2d0 = h1h6d0P2h1 = h1h6Ph1Pd0 = h3h6P
3h1 =h4x14,42 = h6c0P
2c0 = h6d0Ph21 = h6Pc2
0
16 (100) rB23 = mB4
(010) h6P3c0
(001) h1x15,56
18 (1) x18,57
19 (10) g3m = grw = r2m
(01) rR2 = uQ1 = vx′
21 (10) x21,43
(01) Px17,50
22 (010) d20e0gr = d2
0lm = d0e30r = d0e0km = d0e0l
2 = d0g2z = d0gjm =
d0gkl = d0uv = e20gz = e2
0jm = e20kl = e0gim = e0gjl = e0gk2 =
e0u2 = g2rPe0 = g2il = g2jk = riv = rju = m2Pe0 = wPv
(110) d30x
′ = d0e0PQ1 = d0Pd0B21 = d0Pe0Q1 = e0Pd0Q1 =e0Pe0x
′ = gPd0x′ = x10,27P
2e0 = x10,28P2e0 = B23P
2d0
(001) h0x21,43
continued
118 ROBERT R. BRUNER
Stem 95 continued23 (10) h0d
30x
′ = h0d0e0PQ1 = h0d0Pd0B21 = h0d0Pe0Q1 =h0e0Pd0Q1 = h0e0Pe0x
′ = h0gPd0x′ = h0x10,27P
2e0 =h0x10,28P
2e0 = h0B23P2d0 = h1x22,39 = h2d
20PQ1 =
h2d0Pd0Q1 = h2d0Pe0x′ = h2e0Pd0x
′ = h2gP 2Q1 =h2B21P
2e0 = h2x10,27P2d0 = h2x10,28P
2d0 = h4P3Q1 =
c0Px16,35 = d20Ph2Q1 = d0e0Ph2x
′ = d0B2P2e0 =
d0P2h2x10,27 = d0P
2h2x10,28 = e0B2P2d0 = e0P
2h2B21 =gPh2PQ1 = gP 2h2Q1 = Ph2Pd0x10,27 = Ph2Pd0x10,28 =Ph2Pe0B21 = Pc0x16,35 = B2Pd0Pe0 = Q2P
3d0
(01) h20x21,43
24 (01) h30x21,43
(11) P 2x16,35
25 (0011) h0P2x16,35 = h2P
2x16,32 = P 2h2x16,32
(1000) iP 2Q1 = x′P 2j = Q1 P 2i
(1011) d0P2R2 = Pd0PR2 = Pe0R1 = P 2d0R2
(0100) d40u = d3
0ri = d20e0Pv = d2
0gPu = d20jz = d2
0Pd0w = d20Pe0v =
d0e20Pu = d0e0rPj = d0e0iz = d0e0Pd0v = d0e0Pe0u =
d0gPd0u = d0rjPe0 = d0rkPd0 = d0i2m = d0ijl = d0ik
2 =d0j
2k = e20Pd0u = e0gP 2v = e0riPe0 = e0rjPd0 = e0i
2l =e0ijk = e0j
3 = e0wP 2e0 = g2P 2u = griPd0 = gi2k = gij2 =gPe0Pv = gvP 2e0 = gwP 2d0 = gzPj = rlP 2e0 = rmP 2d0 =jmPj = klP j = kPe0z = lPd0z = Pe2
0w
(1100) jx18,20
(0001) h40x21,43
26 (100) h0iP2Q1 = h0jx18,20 = h0x
′P 2j = h0Q1 P 2i = h2ix18,20 =h2x
′P 2i = iP 2h2x′ = B4P
4h2
(001) h50x21,43
(011) h20P
2x16,35 = h0h2P2x16,32 = h0P
2h2x16,32
(111) h0d0P2R2 = h0Pd0PR2 = h0Pe0R1 = h0P
2d0R2 =h2Pd0R1 = d0Ph2R1 = R1P
3e0 = P 3h2P2D1
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 119
Stem 95 continued27 (100) h2
0iP2Q1 = h2
0jx18,20 = h20x
′P 2j = h20Q1 P 2i = h0h2ix18,20 =
h0h2x′P 2i = h0iP
2h2x′ = h0B4P
4h2 = h1Pe0x18,20 =h1x
′P 3e0 = e0Ph1x18,20 = e0x′P 3h1 = Ph1x
′P 2e0 = B1P4e0 =
Pe0P2h1x
′ = x10,28P4h1
(001) h60x21,43
(011) h30P
2x16,35 = h20h2P
2x16,32 = h20P
2h2x16,32 = h31P
2x16,32 =h2
1P2h1x16,32 = h1Ph2
1x16,32 = h1uP 2u = h1X1P4h1 =
h1P3h1W1 = h1Q
2 = h1QPu = h1Pu2 = h3P4Q1 =
Ph1P2h1W1 = Ph1uQ = Ph1uPu = Ph1X1P
3h1 = qP 3u =GP 5h1 = P 2h2
1X1 = P 2h1u2
(111) h20d0P
2R2 = h20Pd0PR2 = h2
0Pe0R1 = h20P
2d0R2 =h0h2Pd0R1 = h0d0Ph2R1 = h0R1P
3e0 = h0P3h2P
2D1 =h1d0P
3Q1 = h1Pd0P2Q1 = h1Q1 P 3d0 = h1P
2d0PQ1 =h2
2P3R2 = h2Ph2P
2R2 = h2P2h2PR2 = h2R1P
3d0 =h2P
3h2R2 = d0Ph1P2Q1 = d0P
2h1PQ1 = d0R1P3h2 =
d0Q1 P 3h1 = Ph1Pd0PQ1 = Ph1Q1 P 2d0 = Ph22PR2 =
Ph2P2h2R2 = Ph2R1P
2d0 = Pd0P2h1Q1 = Pd0P
2h2R1 =x10,27P
4h1
28 (10) d50Pe0 = d4
0e0Pd0 = d30gP 2e0 = d2
0e20P
2e0 = d20e0gP 2d0 =
d20e0Pe2
0 = d20gPd0Pe0 = d0e
30P
2d0 = d0e20Pd0Pe0 = d0e0gPd2
0 =d0g
2P 3e0 = d0iP2v = d0jP
2u = d0uP 2j = d0vP 2i = d0PjPu =e30Pd2
0 = e20gP 3e0 = e0g
2P 3d0 = e0gPe0P2e0 = e0iP
2u =e0uP 2i = g2Pd0P
2e0 = g2Pe0P2d0 = gPe3
0 = riP 2j = rjP 2i =i3j = iPd0Pv = iPe0Pu = iuP 2e0 = ivP 2d0 = jPd0Pu =juP 2d0 = kP 3v = lP 3u = Pd0uPj = wP 3j
(01) h70x21,43
29 (1) h80x21,43
30 (1) h90x21,43
31 (10) d40P
2i = d30iP
2d0 = d20e0P
3j = d20iPd2
0 = d20jP
3e0 =d20kP 3d0 = d2
0Pe0P2j = d2
0PjP 2e0 = d0e0iP3e0 = d0e0jP
3d0 =d0e0Pd0P
2j = d0e0Pe0P2i = d0e0PjP 2d0 = d0giP 3d0 =
d0gPd0P2i = d0iPe0P
2e0 = d0jPd0P2e0 = d0jPe0P
2d0 =d0kPd0P
2d0 = d0lP4e0 = d0mP 4d0 = d0Pd0Pe0Pj =
e20iP
3d0 = e20Pd0P
2i = e0gP 4j = e0iPd0P2e0 = e0iPe0P
2d0 =e0jPd0P
2d0 = e0kP 4e0 = e0lP4d0 = e0Pd2
0Pj = g2P 4i =giPd0P
2d0 = gjP 4e0 = gkP 4d0 = gPe0P3j = gPjP 3e0 =
gP 2e0P2j = iPd0Pe2
0 = jPd20Pe0 = kPd3
0 = kPe0P3e0 =
kP 2e20 = lPd0P
3e0 = lP e0P3d0 = lP 2d0P
2e0 = mPd0P3d0 =
mP 2d20
(01) h100 x21,43
continued
120 ROBERT R. BRUNER
Stem 95 continued32 (10) h1Ph1P
4x′ = h1x′P 5h1 = h1P
3h1x18,20 = Ph1P2h1x18,20 =
Ph1x′P 4h1 = B1P
6h1 = P 2h1x′P 3h1
(01) h110 x21,43
33 (1) h120 x21,43
34 (10) d0iP4j = d0jP
4i = d0P2iP 2j = e0iP
4i = e0P2i2 = rP 6e0 =
i2P 4e0 = ijP 4d0 = iPd0P3j = iP jP 3d0 = iP 2d0P
2j =iP 2e0P
2i = jP 2d0P2i = kP 5j = Pd0PjP 2i = zP 5d0
(01) h130 x21,43
35 (1) h140 x21,43
36 (1) h150 x21,43
37 (10) P 7u
(01) h160 x21,43
38 (1) h170 x21,43
39 (1) h180 x21,43
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 121
Stem 965 (1) h6p
6 (1) h0h6p = h1h6d1
8 (10) x8,83
(01) h1h6q = h2h6r
9 (10) h2x8,78
(01) h0x8,83
10 (1) h20x8,83 = h2
1x8,80
12 (10) x12,64
(01) h3x11,59 = yD2
13 (100) gx9,51 = rx7,40 = A′m = Am
(010) h6P2e0
(001) h0x12,64 = h2x12,60
14 (10) h0h6P2e0 = h2h6P
2d0 = h6d0P2h2 = h6Ph2Pd0
(01) h20x12,64 = h0h2x12,60 = h2
2x12,55 = h5R2 = B22
15 (10) x15,58
(01) h20h6P
2e0 = h0h2h6P2d0 = h0h6d0P
2h2 = h0h6Ph2Pd0
16 (1) h0x15,58
17 (10) h1h6P3c0 = h6c0P
3h1 = h6Ph1P2c0 = h6Pc0P
2h1
(01) h20x15,58 = h2
1x15,56
18 (10) g3t = gnw = grN = nrm = r2t
(01) d0e0B23 = d0gx10,27 = d0gx10,28 = e20x10,27 = e2
0x10,28 = e0gB21 =g2Q1
20 (1) d0x16,38 = e0x16,35 = gx16,32
21 (01) d20P
2D1 = d0e0R2 = d0iB21 = d0jQ1 = d0kx′ = d0Pd0B4 =e0iQ1 = e0jx
′ = gix′ = lPQ1 = Pe0x13,35 = x10,27Pj = x10,28Pj
(11) d0e0gw = d0e0rm = d0g2v = d0grl = e3
0w = e20gv = e2
0rl =e0g
2u = e0grk = g2rj = gmz = jm2 = klm = l3
22 (10) h1x21,43 = h1Px17,50 = Ph1x17,50
(01) h0d20P
2D1 = h0d0e0R2 = h0d0iB21 = h0d0jQ1 = h0d0kx′ =h0d0Pd0B4 = h0e0iQ1 = h0e0jx
′ = h0gix′ = h0lPQ1 =h0Pe0x13,35 = h0x10,27Pj = h0x10,28Pj = h2d
20R2 = h2d0iQ1 =
h2d0jx′ = h2e0ix
′ = h2gPR2 = h2kPQ1 = h2Pd0x13,35 =h2Pe0P
2D1 = h2B4P2e0 = h2B21Pj = h4P
2R2 = d30R1 =
d0f0PQ1 = d0Ph2x13,35 = d0B2Pj = e0Ph2P2D1 = e0Pe0R1 =
e0P2h2B4 = f0Pd0Q1 = f0Pe0x
′ = gPh2R2 = gPd0R1 =Ph2ix10,27 = Ph2ix10,28 = Ph2jB21 = Ph2kQ1 = Ph2lx
′ =Ph2Pe0B4 = D2P
3d0 = iB2Pe0 = jB2Pd0 = Q2P2i =
B5 P 2d0 = PD2P2d0
continued
122 ROBERT R. BRUNER
Stem 96 continued23 (1) h2
0d20P
2D1 = h20d0e0R2 = h2
0d0iB21 = h20d0jQ1 = h2
0d0kx′ =h2
0d0Pd0B4 = h20e0iQ1 = h2
0e0jx′ = h2
0gix′ = h20lPQ1 =
h20Pe0x13,35 = h2
0x10,27Pj = h20x10,28Pj = h0h2d
20R2 =
h0h2d0iQ1 = h0h2d0jx′ = h0h2e0ix
′ = h0h2gPR2 =h0h2kPQ1 = h0h2Pd0x13,35 = h0h2Pe0P
2D1 = h0h2B4P2e0 =
h0h2B21Pj = h0h4P2R2 = h0d
30R1 = h0d0f0PQ1 =
h0d0Ph2x13,35 = h0d0B2Pj = h0e0Ph2P2D1 = h0e0Pe0R1 =
h0e0P2h2B4 = h0f0Pd0Q1 = h0f0Pe0x
′ = h0gPh2R2 =h0gPd0R1 = h0Ph2ix10,27 = h0Ph2ix10,28 = h0Ph2jB21 =h0Ph2kQ1 = h0Ph2lx
′ = h0Ph2Pe0B4 = h0D2P3d0 =
h0iB2Pe0 = h0jB2Pd0 = h0Q2P2i = h0B5 P 2d0 = h0PD2P
2d0 =h1d
30x
′ = h1d0e0PQ1 = h1d0Pd0B21 = h1d0Pe0Q1 =h1e0Pd0Q1 = h1e0Pe0x
′ = h1gPd0x′ = h1x10,27P
2e0 =h1x10,28P
2e0 = h1B23P2d0 = h2
2d0ix′ = h2
2e0PR2 = h22gR1 =
h22jPQ1 = h2
2Pd0P2D1 = h2
2Pe0R2 = h22B4P
2d0 =h2
2Q1 Pj = h2d0Ph2P2D1 = h2d0Pe0R1 = h2d0P
2h2B4 =h2e0Ph2R2 = h2e0Pd0R1 = h2f0Pd0x
′ = h2Ph2iB21 =h2Ph2jQ1 = h2Ph2kx′ = h2Ph2Pd0B4 = h2iB2Pd0 =h4Ph2R1 = c0iR2 = d2
0Ph1B21 = d20B1Pd0 = d0e0Ph1Q1 =
d0e0Ph2R1 = d0f0Ph2x′ = d0gPh1x
′ = d0Ph22B4 = d0Ph2iB2 =
d0P2h1B23 = e2
0Ph1x′ = e0B1P
2e0 = e0P2h1x10,27 =
e0P2h1x10,28 = f0B2P
2d0 = f0P2h2B21 = gB1P
2d0 =gP 2h1B21 = Ph1Pd0B23 = Ph1Pe0x10,27 = Ph1Pe0x10,28 =AP 4h2 = Pc0x16,37 = kB2P
2h2 = B1Pe20 = PAP 3h2
24 (10) iPR2 = jR1
(01) d40g
2 = d30e
20g = d2
0e40 = d2
0iw = d20jv = d2
0ku = d0e0g2Pe0 =
d0e0iv = d0e0ju = d0g3Pd0 = d0giu = d0r
2Pd0 = d0rik =d0rj
2 = d0lPv = d0mPu = d0z2 = e3
0gPe0 = e20g
2Pd0 = e20iu =
e0rij = e0kPv = e0lPu = e0wPj = gri2 = gjPv = gkPu =gvPj = rlP j = rPe0z = ilz = jkz = jPe0w = kPd0w = kPe0v =lPd0v = lP e0u = mPd0u
25 (1) h0iPR2 = h0jR1 = h1P2x16,35 = h2iR1 = Ph1Px16,35 =
P 2h1x16,35 = P 2h2x16,33 = R1P2j
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 123
Stem 96 continued27 (1) d5
0j = d40e0i = d3
0gPj = d30kPe0 = d3
0lPd0 = d20e
20Pj =
d20e0jPe0 = d2
0e0kPd0 = d20giPe0 = d2
0gjPd0 = d20mP 2e0 =
d0e20iPe0 = d0e
20jPd0 = d0e0giPd0 = d0e0lP
2e0 = d0e0mP 2d0 =d0g
2P 2j = d0gkP 2e0 = d0glP 2d0 = d0lP e20 = d0mPd0Pe0 =
e30iPd0 = e2
0gP 2j = e20kP 2e0 = e2
0lP2d0 = e0g
2P 2i = e0gjP 2e0 =e0gkP 2d0 = e0gPe0Pj = e0kPe2
0 = e0lPd0Pe0 = e0mPd20 =
g2iP 2e0 = g2jP 2d0 = g2Pd0Pj = gjPe20 = gkPd0Pe0 = glPd2
0 =gmP 3e0 = rP 3v = i2Pv = ijPu = iuPj = zP 2u
30 (01) d20rP
3d0 = d20i
2Pd0 = d20jP
2j = d20kP 2i = d2
0Pj2 = d0e0iP2j =
d0e0jP2i = d0giP 2i = d0rPd0P
2d0 = d0ijP2e0 = d0ikP 2d0 =
d0iPe0Pj = d0j2P 2d0 = d0jPd0Pj = d0lP
3j = d0zP 3e0 =e20iP
2i = e0rP4e0 = e0i
2P 2e0 = e0ijP2d0 = e0iPd0Pj =
e0kP 3j = e0zP 3d0 = grP 4d0 = gi2P 2d0 = gjP 3j = gPjP 2j =rPd3
0 = rPe0P3e0 = rP 2e2
0 = i2Pe20 = ijPd0Pe0 = ikPd2
0 =ilP 3e0 = imP 3d0 = j2Pd2
0 = jkP 3e0 = jlP 3d0 = k2P 3d0 =kPe0P
2j = kPjP 2e0 = lPd0P2j = lP e0P
2i = lP jP 2d0 =mPd0P
2i = Pd0zP 2e0 = Pe0zP 2d0
(11) P 5Q1
31 (1) h0P5Q1 = Ph2P
4x′ = x′P 5h2 = P 3h2x18,20
32 (1) h20P
5Q1 = h0Ph2P4x′ = h0x
′P 5h2 = h0P3h2x18,20
33 (10) d0P5v = Pd0P
4v = vP 5d0 = P 2d0P3v = PvP 4d0 = P 3d0P
2v
(01) h30P
5Q1 = h20Ph2P
4x′ = h20x
′P 5h2 = h20P
3h2x18,20 =h2
1Ph1P4x′ = h2
1x′P 5h1 = h2
1P3h1x18,20 = h1Ph1P
2h1x18,20 =h1Ph1x
′P 4h1 = h1B1P6h1 = h1P
2h1x′P 3h1 = e0P
4Q =Ph3
1x18,20 = Ph21x
′P 3h1 = Ph1B1P5h1 = Ph1P
2h21x
′ =B1P
2h1P4h1 = B1P
3h21 = QP 4e0
(11) e0P5u = rP 5j = i2P 3j = iP jP 2i = Pe0P
4u = uP 5e0 = zP 4i =P 2e0P
3u = PuP 4e0 = P 3e0P2u
36 (1) d30P
5d0 = d20Pd0P
4d0 = d20P
2d0P3d0 = d0e0P
6e0 = d0gP 6d0 =d0Pd2
0P3d0 = d0Pd0P
2d20 = d0Pe0P
5e0 = d0P2e0P
4e0 =d0P
3e20 = e2
0P6d0 = e0Pd0P
5e0 = e0Pe0P5d0 = e0P
2d0P4e0 =
e0P2e0P
4d0 = e0P3d0P
3e0 = gPd0P5d0 = gP 2d0P
4d0 =gP 3d2
0 = Pd30P
2d0 = Pd0Pe0P4e0 = Pd0P
2e0P3e0 = Pe2
0P4d0 =
Pe0P2d0P
3e0 = Pe0P2e0P
3d0 = P 2d0P2e2
0
38 (1) h1P7u = Ph1P
6u = P 2h1P4Q = P 2h1P
5u = uP 7h1 =P 3h1P
4u = QP 6h1 = PuP 6h1 = P 4h1P3u = P 2uP 5h1
39 (1) d0P7j = iP 7e0 = jP 7d0 = Pd0P
6j = Pe0P6i = PjP 6d0 =
P 2d0P5j = P 2iP 5e0 = P 2jP 5d0 = P 3d0P
4j = P 3e0P4i =
P 3jP 4d0
124 ROBERT R. BRUNER
Stem 977 (10) x7,81
(01) h2h6n
8 (1) h0x7,81
9 (10) h6d0g = h6e20
(01) h20x7,81 = h1x8,83 = h2x8,80
10 (10) x10,76
(01) h0h6d0g = h0h6e20 = h2h6d0e0
11 (10) h0x10,76
(01) h20h6d0g = h2
0h6e20 = h0h2h6d0e0 = h2
2h6d20 = h2h6gPh2 =
h3x10,63 = h4h6P2h2 = h5PD2 = h6c0j = f1Q2
12 (100) gx8,57 = nx7,40 = H1m = tA′ = tA
(010) h6Pj
(001) h20x10,76
13 (010) h0h6Pj = h6Ph2i
(001) h30x10,76 = h3x12,55
(101) h1x12,64 = h22x11,61
14 (01) h40x10,76 = h0h3x12,55 = g2x
′ = xB4
(11) h20h6Pj = h0h6Ph2i = h1h6P
2e0 = h6e0P2h1 = h6Ph1Pe0
15 (10) g2Q2 = rx9,39 = mx8,33
(01) h50x10,76 = h2
0h3x12,55 = h0g2x′ = h0xB4 = rx9,40 = ix8,51
16 (1) h60x10,76 = h3
0h3x12,55 = h20g2x
′ = h20xB4 = h0rx9,40 = h0ix8,51 =
h3xx′ = h3rB4
17 (1) e0gB4 = lB23 = mx10,27 = mx10,28
19 (1) d0x15,41
20 (10) e0g4 = e0gr2 = e0mw = glw = gmv = rlm
(01) d0rx′ = d0iB4 = e0x16,37 = gx16,33 = jx13,35 = kP 2D1 = lR2 =
zQ1
22 (1) Px18,50
23 (10) d30gm = d2
0e20m = d2
0e0gl = d20g
2k = d20ru = d0e
30l = d0e
20gk =
d0e0g2j = d0g
3i = d0r2i = d0vz = e4
0k = e30gj = e2
0g2i =
e0gmPe0 = e0rPv = e0uz = g2lP e0 = g2mPd0 = grPu = rjz =rPd0w = rPe0v = ikw = ilv = imu = j2w = jkv = jlu = k2u
(01) h0Px18,50 = h21x21,43 = h2
1Px17,50 = h1Ph1x17,50 = h2x22,39 =P 2h1x14,42
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 125
Stem 97 continued26 (10) d0x
′P 2d0 = e0P3Q1 = Pd2
0x′ = Pe0P
2Q1 = Q1 P 3e0 =B21P
3d0 = P 2e0PQ1
(01) d40z = d3
0rPe0 = d30il = d3
0jk = d20e0rPd0 = d2
0e0ik = d20e0j
2 =d20gij = d2
0mPj = d0e20ij = d0e0gi2 = d0e0lP j = d0e0Pe0z =
d0grP 2e0 = d0gkPj = d0gPd0z = d0imPe0 = d0jlPe0 =d0jmPd0 = d0k
2Pe0 = d0klPd0 = e30i
2 = e20rP
2e0 = e20kPj =
e20Pd0z = e0grP 2d0 = e0gjPj = e0rPe2
0 = e0ilPe0 = e0imPd0 =e0jkPe0 = e0jlPd0 = e0k
2Pd0 = g2iP j = grPd0Pe0 = gikPe0 =gilPd0 = gj2Pe0 = gjkPd0 = gmP 2j = kmP 2e0 = l2P 2e0 =lmP 2d0 = uP 2v = vP 2u = PuPv
(11) d20x18,20
27 (1) h0d20x18,20 = h0d0x
′P 2d0 = h0e0P3Q1 = h0Pd2
0x′ =
h0Pe0P2Q1 = h0Q1 P 3e0 = h0B21P
3d0 = h0P2e0PQ1 =
h2d0P3Q1 = h2Pd0P
2Q1 = h2Pe0x18,20 = h2x′P 3e0 =
h2Q1 P 3d0 = h2P2d0PQ1 = d0Ph2P
2Q1 = d0P2h2PQ1 =
d0Q1 P 3h2 = e0Ph2x18,20 = e0x′P 3h2 = Ph2Pd0PQ1 =
Ph2x′P 2e0 = Ph2Q1 P 2d0 = B2P
4e0 = Pd0P2h2Q1 =
Pe0P2h2x
′ = x10,27P4h2 = x10,28P
4h2
28 (1) h20d
20x18,20 = h2
0d0x′P 2d0 = h2
0e0P3Q1 = h2
0Pd20x
′ =h2
0Pe0P2Q1 = h2
0Q1 P 3e0 = h20B21P
3d0 = h20P
2e0PQ1 =h0h2d0P
3Q1 = h0h2Pd0P2Q1 = h0h2Pe0x18,20 = h0h2x
′P 3e0 =h0h2Q1 P 3d0 = h0h2P
2d0PQ1 = h0d0Ph2P2Q1 =
h0d0P2h2PQ1 = h0d0Q1 P 3h2 = h0e0Ph2x18,20 = h0e0x
′P 3h2 =h0Ph2Pd0PQ1 = h0Ph2x
′P 2e0 = h0Ph2Q1 P 2d0 = h0B2P4e0 =
h0Pd0P2h2Q1 = h0Pe0P
2h2x′ = h0x10,27P
4h2 = h0x10,28P4h2 =
h22Pd0x18,20 = h2
2x′P 3d0 = h2d0Ph2x18,20 = h2d0x
′P 3h2 =h2Ph2x
′P 2d0 = h2B2P4d0 = h2Pd0P
2h2x′ = h2B21P
4h2 =c0P
3R2 = d0Ph2P2h2x
′ = d0B2P4h2 = Ph2
2Pd0x′ =
Ph2B2P3d0 = Ph2B21P
3h2 = Pc0P2R2 = B2Pd0P
3h2 =B2P
2h2P2d0 = P 2h2
2B21 = P 2c0PR2 = R2 P 3c0
continued
126 ROBERT R. BRUNER
Stem 97 continued29 (10) P 4R2
(01) d30P
2u = d20rP
2i = d20i
3 = d20Pd0Pu = d2
0uP 2d0 = d0e0P3v =
d0gP 3u = d0riP2d0 = d0ijP j = d0Pd2
0u = d0Pe0P2v =
d0vP 3e0 = d0wP 3d0 = d0zP 2j = d0P2e0Pv = e2
0P3u = e0rP
3j =e0i
2Pj = e0Pd0P2v = e0Pe0P
2u = e0uP 3e0 = e0vP 3d0 =e0zP 2i = e0P
2d0Pv = e0P2e0Pu = gPd0P
2u = guP 3d0 =gP 2d0Pu = riPd2
0 = rjP 3e0 = rkP 3d0 = rPe0P2j = rPjP 2e0 =
i2jPe0 = i2kPd0 = ij2Pd0 = ilP 2j = imP 2i = izP 2e0 =jkP 2j = jlP 2i = jzP 2d0 = k2P 2i = kPj2 = Pd0Pe0Pv =Pd0vP 2e0 = Pd0wP 2d0 = Pd0zPj = Pe2
0Pu = Pe0uP 2e0 =Pe0vP 2d0
30 (1) h0P4R2 = Ph2x25,24 = P 3h2R1
31 (1) h20P
4R2 = h0Ph2x25,24 = h0P3h2R1 = h1P
5Q1 = Ph1P4Q1 =
P 2h1P3Q1 = R1P
5h2 = Q1 P 5h1 = P 3h1P2Q1 = PQ1 P 4h1
32 (1) d40P
3e0 = d30e0P
3d0 = d30Pd0P
2e0 = d30Pe0P
2d0 =d20e0Pd0P
2d0 = d20gP 4e0 = d2
0Pd20Pe0 = d0e
20P
4e0 = d0e0gP 4d0 =d0e0Pd3
0 = d0e0Pe0P3e0 = d0e0P
2e20 = d0gPd0P
3e0 =d0gPe0P
3d0 = d0gP 2d0P2e0 = d0Pe2
0P2e0 = e3
0P4d0 =
e20Pd0P
3e0 = e20Pe0P
3d0 = e20P
2d0P2e0 = e0gPd0P
3d0 =e0gP 2d2
0 = e0Pd0Pe0P2e0 = e0Pe2
0P2d0 = g2P 5e0 = gPd2
0P2e0 =
gPd0Pe0P2d0 = iP 4v = jP 4u = Pd0Pe3
0 = uP 4j = vP 4i =PjP 3u = PuP 3j = P 2iP 2v = P 2jP 2u
35 (1) d30P
4i = d20iP
4d0 = d20P
2d0P2i = d0e0P
5j = d0iPd0P3d0 =
d0iP2d2
0 = d0jP5e0 = d0kP 5d0 = d0Pd2
0P2i = d0Pe0P
4j =d0PjP 4e0 = d0P
2e0P3j = d0P
2jP 3e0 = e0iP5e0 = e0jP
5d0 =e0Pd0P
4j = e0Pe0P4i = e0PjP 4d0 = e0P
2d0P3j = e0P
2iP 3e0 =e0P
2jP 3d0 = giP 5d0 = gPd0P4i = gP 2iP 3d0 = iPd2
0P2d0 =
iPe0P4e0 = iP 2e0P
3e0 = jPd0P4e0 = jPe0P
4d0 = jP 2d0P3e0 =
jP 2e0P3d0 = kPd0P
4d0 = kP 2d0P3d0 = lP 6e0 = mP 6d0 =
Pd0Pe0P3j = Pd0PjP 3e0 = Pd0P
2e0P2j = Pe0PjP 3d0 =
Pe0P2d0P
2j = Pe0P2e0P
2i = PjP 2d0P2e0
38 (1) iP 6j = jP 6i = P 2iP 4j = P 2jP 4i
39 (1) h0iP6j = h0jP
6i = h0P2iP 4j = h0P
2jP 4i = h21P
7u =h1Ph1P
6u = h1P2h1P
4Q = h1P2h1P
5u = h1uP 7h1 =h1P
3h1P4u = h1QP 6h1 = h1PuP 6h1 = h1P
4h1P3u =
h1P2uP 5h1 = h2iP
6i = h2P2iP 4i = Ph2
1P4Q = Ph2
1P5u =
Ph1P2h1P
4u = Ph1uP 6h1 = Ph1P3h1P
3u = Ph1QP 5h1 =Ph1PuP 5h1 = Ph1P
4h1P2u = rP 8h2 = qP 8h1 = i2P 6h2 =
iP 2h2P4i = iP 2iP 4h2 = P 2h2
1P3u = P 2h1uP 5h1 =
P 2h1P3h1P
2u = P 2h1QP 4h1 = P 2h1PuP 4h1 = P 2h2P2i2 =
uP 3h1P4h1 = P 3h2
1Q = P 3h21Pu
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 127
Stem 986 (1) h2h6d1 = h4h6g = h5Q3
8 (10) h6m
(01) h2x7,79
9 (10) h3x8,75
(01) h0h6m = h2h6l = h6e0f0
10 (01) h20h6m = h0h2h6l = h0h6e0f0 = h1h6d0g = h1h6e
20 = h2
2h6k =h2h6d0f0
(11) h0h3x8,75 = c2Q2 = f1D2
11 (10) h2x10,73 = d1x7,40 = H1t
(01) h1x10,76
12 (10) h3x11,61
(11) h22x10,65 = h5x11,35
13 (1) h0h3x11,61
14 (100) x14,67
(010) e0gA′ = e0gA = g2D2 = nx9,39 = tx8,33 = lx7,40 = mx7,34
(001) h20h3x11,61 = g2R1 = xX1
(011) rG21
15 (1) h30h3x11,61 = h0g2R1 = h0xX1 = h0rG21 = c0x12,55 = yB4
18 (1) x18,60
19 (1) d0x15,43 = rP 2D1 = uB21 = vQ1 = wx′
21 (1) Px17,52
22 (10) d20g
2r = d20m
2 = d0e20gr = d0e0lm = d0gkm = d0gl2 = d0uw =
d0v2 = e4
0r = e20km = e2
0l2 = e0g
2z = e0gjm = e0gkl = e0uv =g2im = g2jl = g2k2 = gu2 = riw = rjv = rku
(01) d30Q1 = d2
0e0x′ = d0gPQ1 = d0Pd0x10,27 = d0Pd0x10,28 =
d0Pe0B21 = e20PQ1 = e0Pd0B21 = e0Pe0Q1 = gPd0Q1 =
gPe0x′ = B23P
2e0
24 (1) Pd0x16,32
25 (100) iPd0x′ = jP 2Q1 = B4P
3d0 = Q1 P 2j = B21P2i = PjPQ1
(010) d40v = d3
0e0u = d30rj = d2
0e0ri = d20gPv = d2
0kz = d20Pe0w =
d0e20Pv = d0e0gPu = d0e0jz = d0e0Pd0w = d0e0Pe0v =
d0grPj = d0giz = d0gPd0v = d0gPe0u = d0rkPe0 = d0rlPd0 =d0ijm = d0ikl = d0j
2l = d0jk2 = e3
0Pu = e20rPj = e2
0iz =e20Pd0v = e2
0Pe0u = e0gPd0u = e0rjPe0 = e0rkPd0 = e0i2m =
e0ijl = e0ik2 = e0j
2k = g2P 2v = griPe0 = grjPd0 = gi2l =gijk = gj3 = gwP 2e0 = rmP 2e0 = kmPj = l2Pj = lP e0z =mPd0z
(110) kx18,20
(001) h0Pd0x16,32 = h2P2x16,35 = Ph2Px16,35 = P 2h2x16,35
(101) d20R1 = e0P
2R2 = Pe0PR2 = P 2d0P2D1 = P 2e0R2
continued
128 ROBERT R. BRUNER
Stem 98 continued26 (10) h0iPd0x
′ = h0jP2Q1 = h0kx18,20 = h0B4P
3d0 = h0Q1 P 2j =h0B21P
2i = h0PjPQ1 = h2iP2Q1 = h2jx18,20 = h2x
′P 2j =h2Q1 P 2i = f0P
3Q1 = Ph2iPQ1 = Ph2x′Pj = iP 2h2Q1 =
jP 2h2x′ = B2P
3j = P 3h2x13,35
(01) h20Pd0x16,32 = h0h2P
2x16,35 = h0Ph2Px16,35 = h0P2h2x16,35 =
h22P
2x16,32 = h2P2h2x16,32 = Ph2
2x16,32 = P 3h2x13,34
(11) h0d20R1 = h0e0P
2R2 = h0Pe0PR2 = h0P2d0P
2D1 =h0P
2e0R2 = h2d0P2R2 = h2Pd0PR2 = h2Pe0R1 =
h2P2d0R2 = d0Ph2PR2 = d0P
2h2R2 = d0R1P2d0 =
e0Ph2R1 = Ph2Pd0R2 = Pd20R1
27 (1) h20d
20R1 = h2
0e0P2R2 = h2
0iPd0x′ = h2
0jP2Q1 = h2
0kx18,20 =h2
0Pe0PR2 = h20B4P
3d0 = h20Q1 P 2j = h2
0B21P2i = h2
0PjPQ1 =h2
0P2d0P
2D1 = h20P
2e0R2 = h0h2d0P2R2 = h0h2iP
2Q1 =h0h2jx18,20 = h0h2Pd0PR2 = h0h2Pe0R1 = h0h2x
′P 2j =h0h2Q1 P 2i = h0h2P
2d0R2 = h0d0Ph2PR2 = h0d0P2h2R2 =
h0d0R1P2d0 = h0e0Ph2R1 = h0f0P
3Q1 = h0Ph2iPQ1 =h0Ph2Pd0R2 = h0Ph2x
′Pj = h0iP2h2Q1 = h0jP
2h2x′ =
h0B2P3j = h0Pd2
0R1 = h0P3h2x13,35 = h1d
20x18,20 =
h1d0x′P 2d0 = h1e0P
3Q1 = h1Pd20x
′ = h1Pe0P2Q1 =
h1Q1 P 3e0 = h1B21P3d0 = h1P
2e0PQ1 = h22ix18,20 =
h22Pd0R1 = h2
2x′P 2i = h2d0Ph2R1 = h2iP
2h2x′ = h2B4P
4h2 =h2R1P
3e0 = h2P3h2P
2D1 = d20P
2h1x′ = d0Ph1Pd0x
′ =d0B1P
3d0 = d0B21P3h1 = e0Ph1P
2Q1 = e0P2h1PQ1 =
e0R1P3h2 = e0Q1 P 3h1 = f0Ph2x18,20 = f0x
′P 3h2 =gPh1x18,20 = gx′P 3h1 = Ph1Pe0PQ1 = Ph1Q1 P 2e0 =Ph1B21P
2d0 = Ph22ix
′ = Ph2B2P2i = Ph2P
2h2P2D1 =
Ph2B4P3h2 = Ph2R1P
2e0 = iB2P3h2 = B1Pd0P
2d0 =Pd0P
2h1B21 = Pe0P2h1Q1 = Pe0P
2h2R1 = P 2h22B4 = B23P
4h1
28 (1) d70 = d4
0e0Pe0 = d40gPd0 = d3
0e20Pd0 = d2
0e0gP 2e0 = d20g
2P 2d0 =d20gPe2
0 = d20iPu = d0e
30P
2e0 = d0e20gP 2d0 = d0e
20Pe2
0 =d0e0gPd0Pe0 = d0g
2Pd20 = d0iPd0u = d0jP
2v = d0kP 2u =d0vP 2j = d0wP 2i = d0PjPv = e4
0P2d0 = e3
0Pd0Pe0 = e20gPd2
0 =e0g
2P 3e0 = e0iP2v = e0jP
2u = e0uP 2j = e0vP 2i = e0PjPu =g3P 3d0 = g2Pe0P
2e0 = giP 2u = guP 2i = r2P 3d0 = ri2Pd0 =rjP 2j = rkP 2i = rPj2 = i3k = i2j2 = iPe0Pv = ivP 2e0 =iwP 2d0 = izP j = jPd0Pv = jPe0Pu = juP 2e0 = jvP 2d0 =kPd0Pu = kuP 2d0 = lP 3v = mP 3u = Pd0vPj = Pe0uPj
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 129
Stem 98 continued31 (1) d4
0P2j = d3
0e0P2i = d3
0iP2e0 = d3
0jP2d0 = d3
0Pd0Pj =d20e0iP
2d0 = d20gP 3j = d2
0iPd0Pe0 = d20jPd2
0 = d20kP 3e0 =
d20lP
3d0 = d0e20P
3j = d0e0iPd20 = d0e0jP
3e0 = d0e0kP 3d0 =d0e0Pe0P
2j = d0e0PjP 2e0 = d0giP 3e0 = d0gjP 3d0 =d0gPd0P
2j = d0gPe0P2i = d0gPjP 2d0 = d0jPe0P
2e0 =d0kPd0P
2e0 = d0kPe0P2d0 = d0lPd0P
2d0 = d0mP 4e0 =d0Pe2
0Pj = e20iP
3e0 = e20jP
3d0 = e20Pd0P
2j = e20Pe0P
2i =e20PjP 2d0 = e0giP 3d0 = e0gPd0P
2i = e0iPe0P2e0 =
e0jPd0P2e0 = e0jPe0P
2d0 = e0kPd0P2d0 = e0lP
4e0 =e0mP 4d0 = e0Pd0Pe0Pj = g2P 4j = giPd0P
2e0 = giPe0P2d0 =
gjPd0P2d0 = gkP 4e0 = glP 4d0 = gPd2
0Pj = iPe30 = jPd0Pe2
0 =kPd2
0Pe0 = lPd30 = lP e0P
3e0 = lP 2e20 = mPd0P
3e0 =mPe0P
3d0 = mP 2d0P2e0
34 (1) d0rP5d0 = d0i
2P 3d0 = d0iPd0P2i = d0jP
4j = d0kP 4i =d0PjP 3j = d0P
2j2 = e0iP4j = e0jP
4i = e0P2iP 2j = giP 4i =
gP 2i2 = rPd0P4d0 = rP 2d0P
3d0 = i2Pd0P2d0 = ijP 4e0 =
ikP 4d0 = iPe0P3j = iP jP 3e0 = iP 2e0P
2j = j2P 4d0 =jPd0P
3j = jP jP 3d0 = jP 2d0P2j = jP 2e0P
2i = kP 2d0P2i =
lP 5j = Pd0PjP 2j = Pe0PjP 2i = zP 5e0 = Pj2P 2d0
37 (1) P 7v
130 ROBERT R. BRUNER
Stem 995 (1) h3g3 = h4f2
6 (1) h0h3g3 = h0h4f2 = h24p
′
7 (100) x7,83
(010) x7,84
(101) h6t
8 (1) h0x7,84
9 (10) x9,86
(01) h20x7,84 = h2x8,83 = h5G21 = c2D2
10 (10) h22x8,78 = e1A
′
(01) h30x7,84 = h0h2x8,83 = h0h5G21 = h0c2D2 = h1h3x8,75 = pr1 =
D3y = xH1
(11) e1A
11 (1) h3x10,65
12 (10) h21x10,76
(01) h0h3x10,65 = yA′
13 (100) h2x12,64
(001) h20h3x10,65 = h0yA′ = h1h3x11,61 = e1X1 = nG21 = xx8,32
(011) h6d0Pd0
(111) h3x12,58
14 (10) h0h2x12,64 = h22x12,60 = h5P
2D1 = c0x11,61
(01) h0h6d0Pd0 = h2h6P2e0 = h6e0P
2h2 = h6Ph2Pe0
(11) h0h3x12,58
15 (1) h20h3x12,58 = h2
0h6d0Pd0 = h0h2h6P2e0 = h0h6e0P
2h2 =h0h6Ph2Pe0 = h2
2h6P2d0 = h2h6d0P
2h2 = h2h6Ph2Pd0 =h2
3x13,46 = h6d0Ph22 = xx10,27 = yX1
16 (1) h2x15,58
17 (10) gx13,42
(01) Ph1x12,55 = B1x′
18 (1) d0gB23 = e20B23 = e0gx10,27 = e0gx10,28 = g2B21 = uB4
20 (1) d0x16,42 = e0x16,38 = gx16,35
21 (10) d0g2w = d0grm = e2
0gw = e20rm = e0g
2v = e0grl = g3u = g2rk =r2u = km2 = l2m
(01) d20x13,35 = d0e0P
2D1 = d0gR2 = d0ix10,27 = d0ix10,28 =d0jB21 = d0kQ1 = d0lx
′ = d0Pe0B4 = e20R2 = e0iB21 =
e0jQ1 = e0kx′ = e0Pd0B4 = giQ1 = gjx′ = mPQ1 = B23Pj
23 (1) ix16,32
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 131
Stem 99 continued24 (100) i2x′ = B4P
2i
(010) d30e0g
2 = d20e
30g = d2
0rz = d20jw = d2
0kv = d20lu = d0e
50 = d0e0iw =
d0e0jv = d0e0ku = d0g3Pe0 = d0giv = d0gju = d0r
2Pe0 =d0ril = d0rjk = d0mPv = e2
0g2Pe0 = e2
0iv = e20ju = e0g
3Pd0 =e0giu = e0r
2Pd0 = e0rik = e0rj2 = e0lPv = e0mPu = e0z
2 =grij = gkPv = glPu = gwPj = rmPj = imz = jlz = k2z =kPe0w = lPd0w = lP e0v = mPd0v = mPe0u
(110) rx18,20
(001) h0ix16,32
(101) jPR2 = kR1 = Pd0x16,33 = PjR2
25 (10) h0rx18,20 = h0i2x′ = h0B4P
2i = h3P2x16,32
(01) h20ix16,32
(11) h0jPR2 = h0kR1 = h0Pd0x16,33 = h0PjR2 = h1Pd0x16,32 =h2iPR2 = h2jR1 = d0Ph1x16,32 = f0P
2R2 = Ph2iR2 =iPd0R1 = P 2h1x16,38 = P 2h2x16,37 = P 2d0W1
26 (1) h30ix16,32 = h2
0rx18,20 = h20i
2x′ = h20B4P
2i = h0h3P2x16,32 =
h3ix18,20 = h3x′P 2i
27 (10) d50k = d4
0e0j = d40gi = d3
0e20i = d3
0lP e0 = d30mPd0 = d2
0e0gPj =d20e0kPe0 = d2
0e0lPd0 = d20gjPe0 = d2
0gkPd0 = d0e30Pj =
d0e20jPe0 = d0e
20kPd0 = d0e0giPe0 = d0e0gjPd0 = d0e0mP 2e0 =
d0g2iPd0 = d0glP 2e0 = d0gmP 2d0 = d0rP
2u = d0i2u =
d0mPe20 = e3
0iPe0 = e30jPd0 = e2
0giPd0 = e20lP
2e0 = e20mP 2d0 =
e0g2P 2j = e0gkP 2e0 = e0glP 2d0 = e0lP e2
0 = e0mPd0Pe0 =g3P 2i = g2jP 2e0 = g2kP 2d0 = g2Pe0Pj = gkPe2
0 = glPd0Pe0 =gmPd2
0 = rPd0Pu = ruP 2d0 = ijPv = ikPu = ivPj = j2Pu =juPj = zP 2v
(01) h40ix16,32 = h3
0rx18,20 = h30i
2x′ = h30B4P
2i = h20h3P
2x16,32 =h0h3ix18,20 = h0h3x
′P 2i = g2P4i = xiP 2i
(11) r2P 2i = ri3
28 (1) h50ix16,32 = h4
0rx18,20 = h40i
2x′ = h40B4P
2i = h30h3P
2x16,32 =h2
0h3ix18,20 = h20h3x
′P 2i = h0g2P4i = h0xiP 2i = h0r
2P 2i =h0ri
3 = h23P
4x′
continued
132 ROBERT R. BRUNER
Stem 99 continued30 (10) d0P
4x′ = x′P 4d0
(01) d30iP j = d2
0rP3e0 = d2
0i2Pe0 = d2
0ijPd0 = d20kP 2j = d2
0lP2i =
d20zP 2d0 = d0e0rP
3d0 = d0e0i2Pd0 = d0e0jP
2j = d0e0kP 2i =d0e0Pj2 = d0giP 2j = d0gjP 2i = d0rPd0P
2e0 = d0rPe0P2d0 =
d0ikP 2e0 = d0ilP2d0 = d0j
2P 2e0 = d0jkP 2d0 = d0jPe0Pj =d0kPd0Pj = d0mP 3j = d0Pd2
0z = e20iP
2j = e20jP
2i = e0giP 2i =e0rPd0P
2d0 = e0ijP2e0 = e0ikP 2d0 = e0iPe0Pj = e0j
2P 2d0 =e0jPd0Pj = e0lP
3j = e0zP 3e0 = grP 4e0 = gi2P 2e0 = gijP 2d0 =giPd0Pj = gkP 3j = gzP 3d0 = rPd2
0Pe0 = ijPe20 = ikPd0Pe0 =
ilPd20 = imP 3e0 = j2Pd0Pe0 = jkPd2
0 = jlP 3e0 = jmP 3d0 =k2P 3e0 = klP 3d0 = lP e0P
2j = lP jP 2e0 = mPd0P2j =
mPe0P2i = mPjP 2d0 = Pe0zP 2e0
(11) P 2d0x18,20
31 (1) h0d0P4x′ = h0x
′P 4d0 = h0P2d0x18,20 = h2P
5Q1 = Ph2P4Q1 =
P 2h2P3Q1 = Q1 P 5h2 = P 3h2P
2Q1 = PQ1 P 4h2
32 (1) h20d0P
4x′ = h20x
′P 4d0 = h20P
2d0x18,20 = h0h2P5Q1 =
h0Ph2P4Q1 = h0P
2h2P3Q1 = h0Q1 P 5h2 = h0P
3h2P2Q1 =
h0PQ1 P 4h2 = h2Ph2P4x′ = h2x
′P 5h2 = h2P3h2x18,20 =
Ph2P2h2x18,20 = Ph2x
′P 4h2 = B2P6h2 = P 2h2x
′P 3h2
33 (1) d20P
4u = d0rP4i = d0i
2P 2i = d0Pd0P3u = d0uP 4d0 =
d0P2d0P
2u = d0PuP 3d0 = e0P5v = gP 5u = riP 4d0 =
rP 2d0P2i = i3P 2d0 = ijP 3j = iP jP 2j = jP jP 2i = Pd2
0P2u =
Pd0uP 3d0 = Pd0P2d0Pu = Pe0P
4v = uP 2d20 = vP 5e0 =
wP 5d0 = zP 4j = P 2e0P3v = PvP 4e0 = P 3e0P
2v
36 (1) d30P
5e0 = d20e0P
5d0 = d20Pd0P
4e0 = d20Pe0P
4d0 = d20P
2d0P3e0 =
d20P
2e0P3d0 = d0e0Pd0P
4d0 = d0e0P2d0P
3d0 = d0gP 6e0 =d0Pd2
0P3e0 = d0Pd0Pe0P
3d0 = d0Pd0P2d0P
2e0 = d0Pe0P2d2
0 =e20P
6e0 = e0gP 6d0 = e0Pd20P
3d0 = e0Pd0P2d2
0 = e0Pe0P5e0 =
e0P2e0P
4e0 = e0P3e2
0 = gPd0P5e0 = gPe0P
5d0 = gP 2d0P4e0 =
gP 2e0P4d0 = gP 3d0P
3e0 = Pd30P
2e0 = Pd20Pe0P
2d0 =Pe2
0P4e0 = Pe0P
2e0P3e0 = P 2e3
0
39 (1) d20P
6i = d0iP6d0 = d0P
2d0P4i = d0P
2iP 4d0 = e0P7j =
iPd0P5d0 = iP 2d0P
4d0 = iP 3d20 = jP 7e0 = kP 7d0 = Pd2
0P4i =
Pd0P2iP 3d0 = Pe0P
6j = PjP 6e0 = P 2d20P
2i = P 2e0P5j =
P 2jP 5e0 = P 3e0P4j = P 3jP 4e0
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 133
Stem 1004 (1) h4c3
5 (1) h0h4c3 = h5p′
6 (10) h6x
(01) h20h4c3 = h0h5p
′ = h1h3g3 = h1h4f2 = h24p1
7 (1) h0h6x
8 (1000) h2x7,81
(0100) h3h6r
(0010) h1x7,84
(0001) h20h6x
(0101) h1h6t = h1x7,83 = h22h6n = h6c1f0
9 (100) h6e0g
(010) e1H1
(001) h30h6x = h0h3h6r
(011) h3x8,78
10 (100) x10,82
(010) h1x9,86
(001) h40h6x = h2
0h3h6r = h0h3x8,78 = h0h6e0g = h2h6d0g = h2h6e20 =
h23h6i = h4h6Pd0
11 (100) h3x10,67
(010) h0x10,82
(110) h2x10,76
(001) h50h6x = h3
0h3h6r = h20h3x8,78 = h2
0h6e0g = h0h2h6d0g =h0h2h6e
20 = h0h
23h6i = h0h4h6Pd0 = h2
2h6d0e0 = h5PA = h6c0k
(101) H1y
12 (100) h6d0i
(010) h1h3x10,65 = d1G21 = e1x8,32
(001) h20x10,82 = h0h2x10,76
13 (100) x13,73
(010) h0h6d0i = h2h6Pj = h6Ph2j
(001) h30x10,82 = h2
0h2x10,76 = h31x10,76 = h3x12,60 = D3u = Ph1x8,75 =
GB1
14 (1) h20h6d0i = h0h2h6Pj = h0h6Ph2j = h1h3x12,58 = h1h6d0Pd0 =
h2h6Ph2i = h23x12,48 = h6d
20Ph1 = h6f0P
2h2 = h6gP 2h1 =e1x10,27 = g2Q1 = qG21 = yx8,32
16 (1) gx12,44
17 (1) g2B4 = mB23
18 (1) h1Ph1x12,55 = h1B1x′ = uX1
19 (1) e0x15,41
continued
134 ROBERT R. BRUNER
Stem 100 continued20 (10) g5 = g2r2 = gmw = rm2
(01) d0rQ1 = d0jB4 = e0rx′ = e0iB4 = gx16,37 = kx13,35 = lP 2D1 =
mR2 = zB21
22 (1) Px18,55
23 (100) d20e0gm = d2
0g2l = d2
0rv = d0e30m = d0e
20gl = d0e0g
2k = d0e0ru =d0g
3j = d0r2j = d0wz = e4
0l = e30gk = e2
0g2j = e0g
3i = e0r2i =
e0vz = g2mPe0 = grPv = guz = rkz = rPe0w = ilw = imv =jkw = jlv = jmu = k2v = klu
(010) x′Q
(001) x′Pu
(011) rR1 = ix16,33
24 (1) h0rR1 = h0ix16,33 = h0x′Q = i2R1 = X1P
2i
25 (1) h20rR1 = h2
0ix16,33 = h20x
′Q = h0i2R1 = h0X1P
2i = h3iR1
26 (100) d20P
2Q1 = d0Pd0PQ1 = d0x′P 2e0 = d0Q1 P 2d0 = e0x
′P 2d0 =gP 3Q1 = Pd2
0Q1 = Pd0Pe0x′ = B21P
3e0 = x10,27P3d0 =
x10,28P3d0
(010) d50r = d3
0e0z = d30im = d3
0jl = d30k
2 = d20e0rPe0 = d2
0e0il =d20e0jk = d2
0grPd0 = d20gik = d2
0gj2 = d0e20rPd0 = d0e
20ik =
d0e20j
2 = d0e0gij = d0e0mPj = d0g2i2 = d0glP j = d0gPe0z =
d0jmPe0 = d0klPe0 = d0kmPd0 = d0l2Pd0 = d0uPu = e3
0ij =e20gi2 = e2
0lP j = e20Pe0z = e0grP 2e0 = e0gkPj = e0gPd0z =
e0imPe0 = e0jlPe0 = e0jmPd0 = e0k2Pe0 = e0klPd0 =
g2rP 2d0 = g2jP j = grPe20 = gilPe0 = gimPd0 = gjkPe0 =
gjlPd0 = gk2Pd0 = riPu = lmP 2e0 = m2P 2d0 = Pd0u2 =
vP 2v = wP 2u = Pv2
(110) d0e0x18,20
(001) h30rR1 = h3
0ix16,33 = h30x
′Q = h20i
2R1 = h20X1P
2i = h0h3iR1 =h3R1P
2i = riQ
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 135
Stem 100 continued27 (10) h4P
4x′ = c0P2x16,32 = P 2c0x16,32
(01) h40rR1 = h4
0ix16,33 = h40x
′Q = h30i
2R1 = h30X1P
2i = h20h3iR1 =
h0h3R1P2i = h0riQ = h2
3x25,24
(11) h0d20P
2Q1 = h0d0e0x18,20 = h0d0Pd0PQ1 = h0d0x′P 2e0 =
h0d0Q1 P 2d0 = h0e0x′P 2d0 = h0gP 3Q1 = h0Pd2
0Q1 =h0Pd0Pe0x
′ = h0B21P3e0 = h0x10,27P
3d0 = h0x10,28P3d0 =
h2d20x18,20 = h2d0x
′P 2d0 = h2e0P3Q1 = h2Pd2
0x′ =
h2Pe0P2Q1 = h2Q1 P 3e0 = h2B21P
3d0 = h2P2e0PQ1 =
d20P
2h2x′ = d0Ph2Pd0x
′ = d0B2P3d0 = d0B21P
3h2 =e0Ph2P
2Q1 = e0P2h2PQ1 = e0Q1 P 3h2 = gPh2x18,20 =
gx′P 3h2 = Ph2Pe0PQ1 = Ph2Q1 P 2e0 = Ph2B21P2d0 =
B2Pd0P2d0 = Pd0P
2h2B21 = Pe0P2h2Q1 = B23P
4h2
28 (1) h50rR1 = h5
0ix16,33 = h50x
′Q = h40i
2R1 = h40X1P
2i =h3
0h3iR1 = h20h3R1P
2i = h20d
20P
2Q1 = h20d0e0x18,20 =
h20d0Pd0PQ1 = h2
0d0x′P 2e0 = h2
0d0Q1 P 2d0 = h20e0x
′P 2d0 =h2
0gP 3Q1 = h20riQ = h2
0Pd20Q1 = h2
0Pd0Pe0x′ =
h20B21P
3e0 = h20x10,27P
3d0 = h20x10,28P
3d0 = h0h2d20x18,20 =
h0h2d0x′P 2d0 = h0h2e0P
3Q1 = h0h2Pd20x
′ = h0h2Pe0P2Q1 =
h0h2Q1 P 3e0 = h0h2B21P3d0 = h0h2P
2e0PQ1 =h0h
23x25,24 = h0d
20P
2h2x′ = h0d0Ph2Pd0x
′ = h0d0B2P3d0 =
h0d0B21P3h2 = h0e0Ph2P
2Q1 = h0e0P2h2PQ1 =
h0e0Q1 P 3h2 = h0gPh2x18,20 = h0gx′P 3h2 = h0Ph2Pe0PQ1 =h0Ph2Q1 P 2e0 = h0Ph2B21P
2d0 = h0B2Pd0P2d0 =
h0Pd0P2h2B21 = h0Pe0P
2h2Q1 = h0B23P4h2 = h2
2d0P3Q1 =
h22Pd0P
2Q1 = h22Pe0x18,20 = h2
2x′P 3e0 = h2
2Q1 P 3d0 =h2
2P2d0PQ1 = h2d0Ph2P
2Q1 = h2d0P2h2PQ1 =
h2d0Q1 P 3h2 = h2e0Ph2x18,20 = h2e0x′P 3h2 = h2Ph2Pd0PQ1 =
h2Ph2x′P 2e0 = h2Ph2Q1 P 2d0 = h2B2P
4e0 = h2Pd0P2h2Q1 =
h2Pe0P2h2x
′ = h2x10,27P4h2 = h2x10,28P
4h2 = h3i2Q =
c0ix18,20 = c0Pd0R1 = c0x′P 2i = d0Ph2
2PQ1 =d0Ph2P
2h2Q1 = d0Pc0R1 = e0Ph2P2h2x
′ = e0B2P4h2 =
Ph22Pd0Q1 = Ph2
2Pe0x′ = Ph2B2P
3e0 = Ph2x10,27P3h2 =
Ph2x10,28P3h2 = yiP 2i = ix′P 2c0 = B2Pe0P
3h2 =B2P
2h2P2e0 = Q2P
5h2 = P 2h22x10,27 = P 2h2
2x10,28 = B4P4c0 =
P 2D1P3c0
continued
136 ROBERT R. BRUNER
Stem 100 continued29 (10) P 2d0R1
(01) d30P
2v = d20e0P
2u = d20rP
2j = d20i
2j = d20Pd0Pv = d2
0Pe0Pu =d20uP 2e0 = d2
0vP 2d0 = d0e0rP2i = d0e0i
3 = d0e0Pd0Pu =d0e0uP 2d0 = d0gP 3v = d0riP
2e0 = d0rjP2d0 = d0rPd0Pj =
d0ikPj = d0iPd0z = d0j2Pj = d0Pd2
0v = d0Pd0Pe0u =d0wP 3e0 = e2
0P3v = e0gP 3u = e0riP
2d0 = e0ijP j = e0Pd20u =
e0Pe0P2v = e0vP 3e0 = e0wP 3d0 = e0zP 2j = e0P
2e0Pv =grP 3j = gi2Pj = gPd0P
2v = gPe0P2u = guP 3e0 = gvP 3d0 =
gzP 2i = gP 2d0Pv = gP 2e0Pu = riPd0Pe0 = rjPd20 = rkP 3e0 =
rlP 3d0 = i2kPe0 = i2lPd0 = ij2Pe0 = ijkPd0 = imP 2j =j3Pd0 = jlP 2j = jmP 2i = jzP 2e0 = k2P 2j = klP 2i = kzP 2d0 =lP j2 = Pd0wP 2e0 = Pe2
0Pv = Pe0vP 2e0 = Pe0wP 2d0 = Pe0zPj
(11) d0x25,24
30 (1) h0d0x25,24 = h0P2d0R1 = h2P
4R2 = Ph2P3R2 = P 2h2P
2R2 =R1P
4d0 = P 3h2PR2 = R2 P 4h2
31 (1) h20d0x25,24 = h2
0P2d0R1 = h0h2P
4R2 = h0Ph2P3R2 =
h0P2h2P
2R2 = h0R1P4d0 = h0P
3h2PR2 = h0R2 P 4h2 =h1d0P
4x′ = h1x′P 4d0 = h1P
2d0x18,20 = h2Ph2x25,24 =h2P
3h2R1 = d0P2h1x18,20 = d0x
′P 4h1 = Ph1Pd0x18,20 =Ph1x
′P 3d0 = Ph2P2h2R1 = B1P
5d0 = Pd0x′P 3h1 =
P 2h1x′P 2d0 = B21P
5h1
32 (1) d50P
2d0 = d40Pd2
0 = d30e0P
3e0 = d30gP 3d0 = d3
0Pe0P2e0 =
d20e
20P
3d0 = d20e0Pd0P
2e0 = d20e0Pe0P
2d0 = d20gPd0P
2d0 =d20Pd0Pe2
0 = d0e20Pd0P
2d0 = d0e0gP 4e0 = d0e0Pd20Pe0 =
d0g2P 4d0 = d0gPd3
0 = d0gPe0P3e0 = d0gP 2e2
0 = d0iP3u =
d0PuP 2i = e30P
4e0 = e20gP 4d0 = e2
0Pd30 = e2
0Pe0P3e0 = e2
0P2e2
0 =e0gPd0P
3e0 = e0gPe0P3d0 = e0gP 2d0P
2e0 = e0Pe20P
2e0 =g2Pd0P
3d0 = g2P 2d20 = gPd0Pe0P
2e0 = gPe20P
2d0 = iPd0P2u =
iuP 3d0 = iP 2d0Pu = jP 4v = kP 4u = Pd0uP 2i = Pe40 = vP 4j =
wP 4i = PjP 3v = PvP 3j = P 2jP 2v
35 (1) d30P
4j = d20e0P
4i = d20iP
4e0 = d20jP
4d0 = d20Pd0P
3j =d20PjP 3d0 = d2
0P2d0P
2j = d20P
2e0P2i = d0e0iP
4d0 =d0e0P
2d0P2i = d0gP 5j = d0iPd0P
3e0 = d0iPe0P3d0 =
d0iP2d0P
2e0 = d0jPd0P3d0 = d0jP
2d20 = d0kP 5e0 = d0lP
5d0 =d0Pd2
0P2j = d0Pd0Pe0P
2i = d0Pd0PjP 2d0 = e20P
5j =e0iPd0P
3d0 = e0iP2d2
0 = e0jP5e0 = e0kP 5d0 = e0Pd2
0P2i =
e0Pe0P4j = e0PjP 4e0 = e0P
2e0P3j = e0P
2jP 3e0 = giP 5e0 =gjP 5d0 = gPd0P
4j = gPe0P4i = gPjP 4d0 = gP 2d0P
3j =gP 2iP 3e0 = gP 2jP 3d0 = iPd2
0P2e0 = iPd0Pe0P
2d0 =jPd2
0P2d0 = jPe0P
4e0 = jP 2e0P3e0 = kPd0P
4e0 = kPe0P4d0 =
kP 2d0P3e0 = kP 2e0P
3d0 = lPd0P4d0 = lP 2d0P
3d0 = mP 6e0 =Pd3
0Pj = Pe20P
3j = Pe0PjP 3e0 = Pe0P2e0P
2j = PjP 2e20
38 (1) rP 7d0 = i2P 5d0 = iPd0P4i = iP 2iP 3d0 = jP 6j = kP 6i =
Pd0P2i2 = PjP 5j = P 2jP 4j = P 3j2
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 137
Stem 1015 (1) h6e1
7 (010) x7,88
(001) h1h6x = h22h6d1 = h2h4h6g = h2h5Q3 = h3h6n = h6c
21
(101) h6y
8 (10) x8,93
(01) h0h6y
9 (0100) h21x7,84 = h2
2x7,79 = g3Ph1
(1011) h3x8,80
(0010) h0x8,93
(0001) h20h6y = h2h6m = h6f0g
10 (10) h20x8,93
(01) h30h6y = h0h2h6m = h0h6f0g = h1h6e0g = h2
2h6l = h2h6e0f0 =h6c0r
(11) h0h3x8,80 = f1A′
11 (100) g2Q2
(010) x11,80
(001) h30x8,93 = h2
0h3x8,80 = h0f1A′ = h2
1x9,86 = h3pA′ = h3x10,70 =e1x7,33 = xA′′
12 (1) h0g2Q2 = h0x11,80 = pG21
13 (1) h20g2Q2 = h2
0x11,80 = h0pG21 = h21h3x10,65 = h1d1G21 =
h1e1x8,32 = h32x10,65 = h2h5x11,35 = h3d1x8,32 = h3xQ2 =
h4x12,48 = f1X1 = yx7,33
14 (1) g2A′ = g2A = mx7,40
15 (1) h2x14,67
16 (1) Px12,60
17 (01) h0Px12,60 = Ph2x12,55 = B2x′
(11) h6P3d0
18 (10) x18,63
(01) h20Px12,60 = h0h6P
3d0 = h0Ph2x12,55 = h0B2x′ = h5R1
19 (100) g2nr = gtw = gmN = nm2 = rtm
(010) d0x15,47 = e0x15,43 = rx13,35 = ux10,27 = ux10,28 = vB21 =wQ1 = B4z
(001) h0x18,63
20 (1) h20x18,63
21 (10) d0x17,50
(01) h30x18,63
continued
138 ROBERT R. BRUNER
Stem 101 continued22 (010) d3
0B21 = d20e0Q1 = d2
0gx′ = d0e20x
′ = d0Pd0B23 = d0Pe0x10,27 =d0Pe0x10,28 = e0gPQ1 = e0Pd0x10,27 = e0Pd0x10,28 =e0Pe0B21 = gPd0B21 = gPe0Q1
(001) h40x18,63
(101) d0e0g2r = d0e0m
2 = d0glm = d0vw = e30gr = e2
0lm = e0gkm =e0gl2 = e0uw = e0v
2 = g3z = g2jm = g2kl = guv = r2z = rjw =rkv = rlu
23 (1) h50x18,63 = h3x22,39 = R1Q
24 (110) d0Px16,35 = Pd0x16,35
(001) h60x18,63 = h0h3x22,39 = h0R1Q
(101) Pe0x16,32
(011) h1x′Q = h1x
′Pu = Ph1ux′ = qx18,20 = B1P2u = P 2h1x15,42 =
P 2h1x15,43
25 (100) d0iPQ1 = d0x′Pj = iPd0Q1 = iPe0x
′ = jPd0x′ = kP 2Q1 =
B4P3e0 = B21P
2j = x10,28P2i = P 2d0x13,35
(010) d40w = d3
0e0v = d30gu = d3
0rk = d20e
20u = d2
0e0rj = d20gri = d2
0lz =d0e
20ri = d0e0gPv = d0e0kz = d0e0Pe0w = d0g
2Pu = d0gjz =d0gPd0w = d0gPe0v = d0rlPe0 = d0rmPd0 = d0ikm = d0il
2 =d0j
2m = d0jkl = d0k3 = e3
0Pv = e20gPu = e2
0jz = e20Pd0w =
e20Pe0v = e0grPj = e0giz = e0gPd0v = e0gPe0u = e0rkPe0 =
e0rlPd0 = e0ijm = e0ikl = e0j2l = e0jk
2 = g2Pd0u = grjPe0 =grkPd0 = gi2m = gijl = gik2 = gj2k = iu2 = lmPj = mPe0z
(110) lx18,20
(001) h70x18,63 = h2
0h3x22,39 = h20R1Q = h0d0Px16,35 = h0Pd0x16,35 =
h0Pe0x16,32 = h2Pd0x16,32 = d0Ph2x16,32 = P 2h2x16,38 =P 2d0x13,34
(101) d20PR2 = d0e0R1 = d0Pd0R2 = gP 2R2 = x10,27P
2i =P 2e0P
2D1
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 139
Stem 101 continued26 (10) h0d0iPQ1 = h0d0x
′Pj = h0iPd0Q1 = h0iPe0x′ = h0jPd0x
′ =h0kP 2Q1 = h0lx18,20 = h0B4P
3e0 = h0B21P2j = h0x10,28P
2i =h0P
2d0x13,35 = h2iPd0x′ = h2jP
2Q1 = h2kx18,20 = h2B4P3d0 =
h2Q1 P 2j = h2B21P2i = h2PjPQ1 = h4x25,24 = d0f0x18,20 =
d0Ph2ix′ = d0B2P
2i = d0B4P3h2 = f0x
′P 2d0 = Ph2jPQ1 =Ph2B4P
2d0 = Ph2Q1 Pj = iB2P2d0 = iP 2h2B21 = jP 2h2Q1 =
kP 2h2x′ = Pd0P
2h2B4
(01) h80x18,63 = h3
0h3x22,39 = h30R1Q = h2
0d0Px16,35 = h20Pd0x16,35 =
h20Pe0x16,32 = h0h2Pd0x16,32 = h0d0Ph2x16,32 = h0P
2h2x16,38 =h0P
2d0x13,34 = h22P
2x16,35 = h2Ph2Px16,35 = h2P2h2x16,35 =
Ph22x16,35
(11) h0d20PR2 = h0d0e0R1 = h0d0Pd0R2 = h0gP 2R2 =
h0x10,27P2i = h0P
2e0P2D1 = h2d
20R1 = h2e0P
2R2 =h2Pe0PR2 = h2P
2d0P2D1 = h2P
2e0R2 = d0P2h2P
2D1 =d0R1P
2e0 = e0Ph2PR2 = e0P2h2R2 = e0R1P
2d0 = gPh2R1 =Ph2Pd0P
2D1 = Ph2Pe0R2 = Pd0Pe0R1
27 (1) h20d
20PR2 = h2
0d0e0R1 = h20d0iPQ1 = h2
0d0Pd0R2 =h2
0d0x′Pj = h2
0gP 2R2 = h20iPd0Q1 = h2
0iPe0x′ =
h20jPd0x
′ = h20kP 2Q1 = h2
0lx18,20 = h20B4P
3e0 =h2
0B21P2j = h2
0x10,27P2i = h2
0x10,28P2i = h2
0P2d0x13,35 =
h20P
2e0P2D1 = h0h2d
20R1 = h0h2e0P
2R2 = h0h2iPd0x′ =
h0h2jP2Q1 = h0h2kx18,20 = h0h2Pe0PR2 = h0h2B4P
3d0 =h0h2Q1 P 2j = h0h2B21P
2i = h0h2PjPQ1 = h0h2P2d0P
2D1 =h0h2P
2e0R2 = h0h4x25,24 = h0d0f0x18,20 = h0d0Ph2ix′ =
h0d0B2P2i = h0d0P
2h2P2D1 = h0d0B4P
3h2 = h0d0R1P2e0 =
h0e0Ph2PR2 = h0e0P2h2R2 = h0e0R1P
2d0 = h0f0x′P 2d0 =
h0gPh2R1 = h0Ph2jPQ1 = h0Ph2Pd0P2D1 = h0Ph2Pe0R2 =
h0Ph2B4P2d0 = h0Ph2Q1 Pj = h0iB2P
2d0 = h0iP2h2B21 =
h0jP2h2Q1 = h0kP 2h2x
′ = h0Pd0Pe0R1 = h0Pd0P2h2B4 =
h1d20P
2Q1 = h1d0e0x18,20 = h1d0Pd0PQ1 = h1d0x′P 2e0 =
h1d0Q1 P 2d0 = h1e0x′P 2d0 = h1gP 3Q1 = h1Pd2
0Q1 =h1Pd0Pe0x
′ = h1B21P3e0 = h1x10,27P
3d0 = h1x10,28P3d0 =
h22d0P
2R2 = h22iP
2Q1 = h22jx18,20 = h2
2Pd0PR2 = h22Pe0R1 =
h22x
′P 2j = h22Q1 P 2i = h2
2P2d0R2 = h2d0Ph2PR2 =
h2d0P2h2R2 = h2d0R1P
2d0 = h2e0Ph2R1 = h2f0P3Q1 =
h2Ph2iPQ1 = h2Ph2Pd0R2 = h2Ph2x′Pj = h2iP
2h2Q1 =h2jP
2h2x′ = h2B2P
3j = h2Pd20R1 = h2P
3h2x13,35 = c0iR1 =d20Ph1PQ1 = d2
0P2h1Q1 = d2
0P2h2R1 = d0e0P
2h1x′ =
d0Ph1Pd0Q1 = d0Ph1Pe0x′ = d0Ph2
2R2 = d0Ph2Pd0R1 =d0B1P
3e0 = d0x10,27P3h1 = d0x10,28P
3h1 = e0Ph1Pd0x′ =
(continued)
140 ROBERT R. BRUNER
Stem 101 continued27 (1) (continued) = e0B1P
3d0 = e0B21P3h1 = f0Ph2P
2Q1 =f0P
2h2PQ1 = f0Q1 P 3h2 = gPh1P2Q1 = gP 2h1PQ1 =
gR1P3h2 = gQ1 P 3h1 = Ph1B21P
2e0 = Ph1x10,27P2d0 =
Ph1x10,28P2d0 = Ph2
2iQ1 = Ph22jx
′ = Ph2B2P2j =
Ph2P2h2x13,35 = D2P
5h2 = jB2P3h2 = B1Pd0P
2e0 =B1Pe0P
2d0 = B2P2h2Pj = Pd0P
2h1x10,27 = Pd0P2h1x10,28 =
Pe0P2h1B21 = B5 P 4h2 = PD2P
4h2 = P 2c0x16,33
28 (1) d60e0 = d4
0gPe0 = d30e
20Pe0 = d3
0e0gPd0 = d20e
30Pd0 = d2
0g2P 2e0 =
d20iPv = d2
0jPu = d20uPj = d0e
20gP 2e0 = d0e0g
2P 2d0 =d0e0gPe2
0 = d0e0iPu = d0g2Pd0Pe0 = d0riP j = d0i
2z =d0iPd0v = d0iPe0u = d0jPd0u = d0kP 2v = d0lP
2u = d0wP 2j =e40P
2e0 = e30gP 2d0 = e3
0Pe20 = e2
0gPd0Pe0 = e0g2Pd2
0 =e0iPd0u = e0jP
2v = e0kP 2u = e0vP 2j = e0wP 2i = e0PjPv =g3P 3e0 = giP 2v = gjP 2u = guP 2j = gvP 2i = gPjPu =r2P 3e0 = ri2Pe0 = rijPd0 = rkP 2j = rlP 2i = rzP 2d0 = i3l =i2jk = ij3 = iwP 2e0 = jPe0Pv = jvP 2e0 = jwP 2d0 = jzPj =kPd0Pv = kPe0Pu = kuP 2e0 = kvP 2d0 = lPd0Pu = luP 2d0 =mP 3v = Pd0wPj = Pe0vPj
29 (1) Ph1P2x16,32 = P 3h1x16,32
31 (1) d40iPd0 = d3
0e0P2j = d3
0gP 2i = d30jP
2e0 = d30kP 2d0 =
d30Pe0Pj = d2
0e20P
2i = d20e0iP
2e0 = d20e0jP
2d0 = d20e0Pd0Pj =
d20giP 2d0 = d2
0iPe20 = d2
0jPd0Pe0 = d20kPd2
0 = d20lP
3e0 =d20mP 3d0 = d0e
20iP
2d0 = d0e0gP 3j = d0e0iPd0Pe0 = d0e0jPd20 =
d0e0kP 3e0 = d0e0lP3d0 = d0giPd2
0 = d0gjP 3e0 = d0gkP 3d0 =d0gPe0P
2j = d0gPjP 2e0 = d0kPe0P2e0 = d0lPd0P
2e0 =d0lP e0P
2d0 = d0mPd0P2d0 = e3
0P3j = e2
0iPd20 = e2
0jP3e0 =
e20kP 3d0 = e2
0Pe0P2j = e2
0PjP 2e0 = e0giP 3e0 = e0gjP 3d0 =e0gPd0P
2j = e0gPe0P2i = e0gPjP 2d0 = e0jPe0P
2e0 =e0kPd0P
2e0 = e0kPe0P2d0 = e0lPd0P
2d0 = e0mP 4e0 =e0Pe2
0Pj = g2iP 3d0 = g2Pd0P2i = giPe0P
2e0 = gjPd0P2e0 =
gjPe0P2d0 = gkPd0P
2d0 = glP 4e0 = gmP 4d0 = gPd0Pe0Pj =rP 4u = i2P 2u = iuP 2i = jPe3
0 = kPd0Pe20 = lPd2
0Pe0 =mPd3
0 = mPe0P3e0 = mP 2e2
0
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 141
Stem 101 continued34 (01) d2
0iP3j = d2
0PjP 2i = d0rP5e0 = d0i
2P 3e0 = d0ijP3d0 =
d0iPd0P2j = d0iPe0P
2i = d0iP jP 2d0 = d0jPd0P2i = d0kP 4j =
d0lP4i = d0zP 4d0 = e0rP
5d0 = e0i2P 3d0 = e0iPd0P
2i =e0jP
4j = e0kP 4i = e0PjP 3j = e0P2j2 = giP 4j = gjP 4i =
gP 2iP 2j = rPd0P4e0 = rPe0P
4d0 = rP 2d0P3e0 = rP 2e0P
3d0 =i2Pd0P
2e0 = i2Pe0P2d0 = ijPd0P
2d0 = ikP 4e0 = ilP 4d0 =iPd2
0Pj = j2P 4e0 = jkP 4d0 = jPe0P3j = jP jP 3e0 =
jP 2e0P2j = kPd0P
3j = kPjP 3d0 = kP 2d0P2j = kP 2e0P
2i =lP 2d0P
2i = mP 5j = Pd0zP 3d0 = Pe0PjP 2j = zP 2d20 =
Pj2P 2e0
(11) P 4x18,20
35 (1) h0P4x18,20
36 (1) h20P
4x18,20
37 (10) d0P6u = Pd0P
5u = uP 6d0 = P 2d0P4u = PuP 5d0 = P 3d0P
3u =P 2uP 4d0
(01) h30P
4x18,20
(11) rP 6i = i2P 4i = iP 2i2
38 (1) h40P
4x18,20 = h0rP6i = h0i
2P 4i = h0iP2i2
39 (1) h50P
4x18,20 = h20rP
6i = h20i
2P 4i = h20iP
2i2 = h3iP6i = h3P
2iP 4i
142 ROBERT R. BRUNER
Stem 1026 (1) h1h6e1 = h3h6d1
8 (100) h3x7,79
(010) h1h6y = h2h6t = h3h6q = h6c1g
(001) h1x7,88 = h2x7,84
(011) h2x7,83 = h4x7,74
9 (1) h0h3x7,79 = h1x8,93 = c2A′ = f1H1
10 (100) h6u
(010) g2D2
(001) h1h3x8,80 = h2x9,86
11 (1) h0g2D2 = h3x10,73
12 (1) h20g2D2 = h0h3x10,73 = h1g2Q2 = h3e1Q2 = h3xD2 = c2X1 =
f1x8,32 = yA′′
13 (01) h6d0Pe0 = h6e0Pd0
(11) g2H1 = tx7,40
14 (10) x14,74
(01) h0h6d0Pe0 = h0h6e0Pd0 = h22x12,64 = h2h6d0Pd0 = h5x13,34 =
h6d20Ph2 = h6gP 2h2
15 (1) x15,65
16 (100) g2x8,33 = mx9,39 = Q2w
(010) h6P2i
(001) h0x15,65 = Ph2x11,61
17 (10) h0h6P2i
(01) h20x15,65 = h0Ph2x11,61 = h1Px12,60 = Ph1x12,60 = B1Q1 =
B2R1
18 (10) e0gB23 = g2x10,27 = g2x10,28 = vB4
(01) h20h6P
2i = h1h6P3d0 = h4x17,50 = h6d0P
3h1 = h6Ph1P2d0 =
h6Pd0P2h1 = c0x15,56
19 (10) x19,58
(01) h1x18,63
20 (10) d0x16,48 = e0x16,42 = gx16,38
(01) h0x19,58
21 (100) e0g2w = e0grm = g3v = g2rl = r2v = lm2
(010) d30B4 = d0e0x13,35 = d0gP 2D1 = d0iB23 = d0jx10,27 =
d0jx10,28 = d0kB21 = d0lQ1 = d0mx′ = e20P
2D1 = e0gR2 =e0ix10,27 = e0ix10,28 = e0jB21 = e0kQ1 = e0lx
′ = e0Pe0B4 =giB21 = gjQ1 = gkx′ = gPd0B4
(001) h20x19,58
22 (1) h30x19,58 = h3x21,43
23 (01) h40x19,58 = h0h3x21,43
(11) ix16,35 = jx16,32
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 143
Stem 102 continued24 (100) d3
0g3 = d3
0r2 = d2
0e20g
2 = d20kw = d2
0lv = d20mu = d0e
40g = d0e0rz =
d0e0jw = d0e0kv = d0e0lu = d0giw = d0gjv = d0gku = d0rim =d0rjl = d0rk
2 = e60 = e2
0iw = e20jv = e2
0ku = e0g3Pe0 = e0giv =
e0gju = e0r2Pe0 = e0ril = e0rjk = e0mPv = g4Pd0 = g2iu =
gr2Pd0 = grik = grj2 = glPv = gmPu = gz2 = jmz = klz =lP e0w = mPd0w = mPe0v
(010) rP 2Q1 = i2Q1 = ijx′ = B4P2j
(001) h50x19,58 = h2
0h3x21,43 = h0ix16,35 = h0jx16,32 = h2ix16,32 =P 2h2x15,41
(011) d0iR2 = kPR2 = lR1 = Pd0x16,37 = Pe0x16,33 = PjP 2D1
25 (10) h0rP2Q1 = h0i
2Q1 = h0ijx′ = h0B4P
2j = h21x
′Q =h2
1x′Pu = h1Ph1ux′ = h1qx18,20 = h1B1P
2u = h1P2h1x15,42 =
h1P2h1x15,43 = h2rx18,20 = h2i
2x′ = h2B4P2i = h3P
2x16,35 =D3P
5h1 = Ph21x15,42 = Ph2
1x15,43 = Ph1B1Q = Ph1B1Pu =rP 2h2x
′ = qP 2h1x′ = iP 2h2B4 = B1P
2h1u = PD3P4h1 =
X1P3e0 = P 2D3P
3h1
(01) h60x19,58 = h3
0h3x21,43 = h20ix16,35 = h2
0jx16,32 = h0h2ix16,32 =h0P
2h2x15,41 = h1Pe0x16,32 = e0Ph1x16,32 = P 2e0W1
(11) h0d0iR2 = h0kPR2 = h0lR1 = h0Pd0x16,37 = h0Pe0x16,33 =h0PjP 2D1 = h1d0Px16,35 = h1Pd0x16,35 = h2jPR2 = h2kR1 =h2Pd0x16,33 = h2PjR2 = c0x22,39 = d0f0R1 = d0Ph1x16,35 =d0Ph2x16,33 = d0R1Pj = Ph2iP
2D1 = Ph2jR2 = iPe0R1 =jPd0R1 = P 2h1x16,42
27 (1) d50l = d4
0e0k = d40gj = d3
0e20j = d3
0e0gi = d30mPe0 = d2
0e30i =
d20e0lP e0 = d2
0e0mPd0 = d20g
2Pj = d20gkPe0 = d2
0glPd0 =d0e
20gPj = d0e
20kPe0 = d0e
20lPd0 = d0e0gjPe0 = d0e0gkPd0 =
d0g2iPe0 = d0g
2jPd0 = d0gmP 2e0 = d0rP2v = d0i
2v = d0iju =d0zPu = e4
0Pj = e30jPe0 = e3
0kPd0 = e20giPe0 = e2
0gjPd0 =e20mP 2e0 = e0g
2iPd0 = e0glP 2e0 = e0gmP 2d0 = e0rP2u =
e0i2u = e0mPe2
0 = g3P 2j = g2kP 2e0 = g2lP 2d0 = glPe20 =
gmPd0Pe0 = r2P 2j = ri2j = rPd0Pv = rPe0Pu = ruP 2e0 =rvP 2d0 = ikPv = ilPu = iwPj = j2Pv = jkPu = jvPj =kuPj = Pd0uz
continued
144 ROBERT R. BRUNER
Stem 102 continued30 (100) d0P
4Q1 = Pd0P3Q1 = Q1 P 4d0 = P 2d0P
2Q1 = PQ1 P 3d0
(010) uP 3u = PuP 2u
(110) P 2e0x18,20
(001) h1Ph1P2x16,32 = h1P
3h1x16,32 = Ph1P2h1x16,32 = X1P
5h1 =QP 2u = W1P
4h1
(101) e0P4x′ = x′P 4e0
(011) d40i
2 = d30rP
2d0 = d30jP j = d2
0e0iP j = d20rPd2
0 = d20ijPe0 =
d20ikPd0 = d2
0j2Pd0 = d2
0lP2j = d2
0mP 2i = d20zP 2e0 =
d0e0rP3e0 = d0e0i
2Pe0 = d0e0ijPd0 = d0e0kP 2j = d0e0lP2i =
d0e0zP 2d0 = d0grP 3d0 = d0gi2Pd0 = d0gjP 2j = d0gkP 2i =d0gPj2 = d0rPe0P
2e0 = d0ilP2e0 = d0imP 2d0 = d0jkP 2e0 =
d0jlP2d0 = d0k
2P 2d0 = d0kPe0Pj = d0lPd0Pj = d0Pd0Pe0z =e20rP
3d0 = e20i
2Pd0 = e20jP
2j = e20kP 2i = e2
0Pj2 = e0giP 2j =e0gjP 2i = e0rPd0P
2e0 = e0rPe0P2d0 = e0ikP 2e0 = e0ilP
2d0 =e0j
2P 2e0 = e0jkP 2d0 = e0jPe0Pj = e0kPd0Pj = e0mP 3j =e0Pd2
0z = g2iP 2i = grPd0P2d0 = gijP 2e0 = gikP 2d0 =
giPe0Pj = gj2P 2d0 = gjPd0Pj = glP 3j = gzP 3e0 = rPd0Pe20 =
ikPe20 = ilPd0Pe0 = imPd2
0 = j2Pe20 = jkPd0Pe0 = jlPd2
0 =jmP 3e0 = k2Pd2
0 = klP 3e0 = kmP 3d0 = l2P 3d0 = mPe0P2j =
mPjP 2e0
31 (1) h0d0P4Q1 = h0e0P
4x′ = h0Pd0P3Q1 = h0x
′P 4e0 =h0Q1 P 4d0 = h0P
2d0P2Q1 = h0P
2e0x18,20 = h0PQ1 P 3d0 =h2d0P
4x′ = h2x′P 4d0 = h2P
2d0x18,20 = d0P2h2x18,20 =
d0x′P 4h2 = Ph2Pd0x18,20 = Ph2x
′P 3d0 = B2P5d0 =
Pd0x′P 3h2 = P 2h2x
′P 2d0 = B21P5h2
32 (1) h20d0P
4Q1 = h20e0P
4x′ = h20Pd0P
3Q1 = h20x
′P 4e0 =h2
0Q1 P 4d0 = h20P
2d0P2Q1 = h2
0P2e0x18,20 = h2
0PQ1 P 3d0 =h0h2d0P
4x′ = h0h2x′P 4d0 = h0h2P
2d0x18,20 = h0d0P2h2x18,20 =
h0d0x′P 4h2 = h0Ph2Pd0x18,20 = h0Ph2x
′P 3d0 = h0B2P5d0 =
h0Pd0x′P 3h2 = h0P
2h2x′P 2d0 = h0B21P
5h2 = h22P
5Q1 =h2Ph2P
4Q1 = h2P2h2P
3Q1 = h2Q1 P 5h2 = h2P3h2P
2Q1 =h2PQ1 P 4h2 = Ph2
2P3Q1 = Ph2P
2h2P2Q1 = Ph2Q1 P 4h2 =
Ph2P3h2PQ1 = Pc0x25,24 = P 2h2
2PQ1 = P 2h2Q1 P 3h2 =P 3c0R1
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 145
Stem 102 continued33 (01) d2
0P4v = d0e0P
4u = d0rP4j = d0i
2P 2j = d0ijP2i = d0Pd0P
3v =d0Pe0P
3u = d0uP 4e0 = d0vP 4d0 = d0P2d0P
2v = d0P2e0P
2u =d0PuP 3e0 = d0PvP 3d0 = e0rP
4i = e0i2P 2i = e0Pd0P
3u =e0uP 4d0 = e0P
2d0P2u = e0PuP 3d0 = gP 5v = riP 4e0 =
rjP 4d0 = rPd0P3j = rPjP 3d0 = rP 2d0P
2j = rP 2e0P2i =
i3P 2e0 = i2jP 2d0 = i2Pd0Pj = ikP 3j = izP 3d0 = j2P 3j =jP jP 2j = kPjP 2i = Pd2
0P2v = Pd0Pe0P
2u = Pd0uP 3e0 =Pd0vP 3d0 = Pd0zP 2i = Pd0P
2d0Pv = Pd0P2e0Pu =
Pe0uP 3d0 = Pe0P2d0Pu = uP 2d0P
2e0 = vP 2d20 = wP 5e0 = Pj3
(11) P 4R1
34 (1) h0P4R1
35 (10) h1P4x18,20 = P 2h1P
4x′ = x′P 6h1 = P 4h1x18,20
(01) h20P
4R1
36 (10) d40P
4d0 = d30Pd0P
3d0 = d30P
2d20 = d2
0e0P5e0 = d2
0gP 5d0 =d20Pd2
0P2d0 = d2
0Pe0P4e0 = d2
0P2e0P
3e0 = d0e20P
5d0 =d0e0Pd0P
4e0 = d0e0Pe0P4d0 = d0e0P
2d0P3e0 =
d0e0P2e0P
3d0 = d0gPd0P4d0 = d0gP 2d0P
3d0 = d0Pd40 =
d0Pd0Pe0P3e0 = d0Pd0P
2e20 = d0Pe2
0P3d0 = d0Pe0P
2d0P2e0 =
e20Pd0P
4d0 = e20P
2d0P3d0 = e0gP 6e0 = e0Pd2
0P3e0 =
e0Pd0Pe0P3d0 = e0Pd0P
2d0P2e0 = e0Pe0P
2d20 = g2P 6d0 =
gPd20P
3d0 = gPd0P2d2
0 = gPe0P5e0 = gP 2e0P
4e0 = gP 3e20 =
iP 5u = Pd20Pe0P
2e0 = Pd0Pe20P
2d0 = PuP 4i = P 2iP 3u
(01) h30P
4R1 = iP 4Q = QP 4i
37 (1) h40P
4R1 = h0iP4Q = h0QP 4i
38 (1) h50P
4R1 = h20iP
4Q = h20QP 4i
146 ROBERT R. BRUNER
Stem 1035 (1) h6f1
6 (1) h0h6f1 = h3h6p
7 (10) x7,90
(01) h20h6f1 = h0h3h6p = h2
1h6e1 = h1h3h6d1
9 (1) h3x8,83
10 (1) h0h3x8,83
11 (1) h1h6u
12 (10) x12,78
(01) h6d0j = h6e0i
13 (1) h0h6d0j = h0h6e0i = h0x12,78 = h2h6d0i = h6f0Pd0 = h6Ph2k
14 (1) h20h6d0j = h2
0h6e0i = h20x12,78 = h0h2h6d0i = h0h6f0Pd0 =
h0h6Ph2k = h1h6d0Pe0 = h1h6e0Pd0 = h22h6Pj = h2h6Ph2j =
h6d0e0Ph1 = B1Q2
15 (10) e0gx7,40 = g2x7,34 = tx9,39 = D2w = A′v = Av = mG21 = Q2N
(01) h1x14,74
19 (1) gx15,41
20 (10) d0rB21 = d0kB4 = e0rQ1 = e0jB4 = grx′ = giB4 = lx13,35 =mP 2D1 = zx10,27 = zx10,28
(01) h21x18,63 = Ph1x15,56
22 (1) d0x18,50
23 (10) d20g
2m = d20rw = d0e
20gm = d0e0g
2l = d0e0rv = d0g3k = d0gru =
d0r2k = e4
0m = e30gl = e2
0g2k = e2
0ru = e0g3j = e0r
2j = e0wz =g4i = gr2i = gvz = rlz = imw = jlw = jmv = k2w = klv =kmu = l2u
(01) rPR2 = ix16,37 = jx16,33 = uPQ1 = x′Pv = Q1 Pu
25 (1) P 2x17,50
26 (10) d20Pd0x
′ = d0e0P2Q1 = d0Pe0PQ1 = d0Q1 P 2e0 = d0B21P
2d0 =e0Pd0PQ1 = e0x
′P 2e0 = e0Q1 P 2d0 = gx′P 2d0 = Pd20B21 =
Pd0Pe0Q1 = Pe20x
′ = x10,27P3e0 = x10,28P
3e0 = B23P3d0
(01) d40e0r = d3
0gz = d30jm = d3
0kl = d20e
20z = d2
0e0im = d20e0jl =
d20e0k
2 = d20grPe0 = d2
0gil = d20gjk = d0e
20rPe0 = d0e
20il =
d0e20jk = d0e0grPd0 = d0e0gik = d0e0gj2 = d0g
2ij = d0gmPj =d0kmPe0 = d0l
2Pe0 = d0lmPd0 = d0uPv = d0vPu = e30rPd0 =
e30ik = e3
0j2 = e2
0gij = e20mPj = e0g
2i2 = e0glP j = e0gPe0z =e0jmPe0 = e0klPe0 = e0kmPd0 = e0l
2Pd0 = e0uPu =g2rP 2e0 = g2kPj = g2Pd0z = gimPe0 = gjlPe0 = gjmPd0 =gk2Pe0 = gklPd0 = riPv = rjPu = ruPj = iuz = m2P 2e0 =Pd0uv = Pe0u
2 = wP 2v
(11) d0gx18,20 = e20x18,20
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 147
Stem 103 continued27 (1) h0d
20Pd0x
′ = h0d0e0P2Q1 = h0d0gx18,20 = h0d0Pe0PQ1 =
h0d0Q1 P 2e0 = h0d0B21P2d0 = h0e
20x18,20 = h0e0Pd0PQ1 =
h0e0x′P 2e0 = h0e0Q1 P 2d0 = h0gx′P 2d0 = h0Pd2
0B21 =h0Pd0Pe0Q1 = h0Pe2
0x′ = h0x10,27P
3e0 = h0x10,28P3e0 =
h0B23P3d0 = h2d
20P
2Q1 = h2d0e0x18,20 = h2d0Pd0PQ1 =h2d0x
′P 2e0 = h2d0Q1 P 2d0 = h2e0x′P 2d0 = h2gP 3Q1 =
h2Pd20Q1 = h2Pd0Pe0x
′ = h2B21P3e0 = h2x10,27P
3d0 =h2x10,28P
3d0 = h4P4Q1 = c0P
2x16,35 = d20Ph2PQ1 =
d20P
2h2Q1 = d0e0P2h2x
′ = d0Ph2Pd0Q1 = d0Ph2Pe0x′ =
d0B2P3e0 = d0x10,27P
3h2 = d0x10,28P3h2 = e0Ph2Pd0x
′ =e0B2P
3d0 = e0B21P3h2 = gPh2P
2Q1 = gP 2h2PQ1 =gQ1 P 3h2 = Ph1x22,39 = Ph2B21P
2e0 = Ph2x10,27P2d0 =
Ph2x10,28P2d0 = Pc0Px16,35 = B2Pd0P
2e0 = B2Pe0P2d0 =
Q2P4d0 = Pd0P
2h2x10,27 = Pd0P2h2x10,28 = Pe0P
2h2B21 =P 2c0x16,35
28 (1) P 3x16,35
29 (100) Pjx18,20
(010) d40Pu = d3
0Pd0u = d20e0P
2v = d20gP 2u = d2
0riPd0 = d20i
2k =d20ij
2 = d20Pe0Pv = d2
0vP 2e0 = d20wP 2d0 = d2
0zPj = d0e20P
2u =d0e0rP
2j = d0e0i2j = d0e0Pd0Pv = d0e0Pe0Pu = d0e0uP 2e0 =
d0e0vP 2d0 = d0grP 2i = d0gi3 = d0gPd0Pu = d0guP 2d0 =d0rjP
2e0 = d0rkP 2d0 = d0rPe0Pj = d0ilP j = d0iPe0z =d0jkPj = d0jPd0z = d0Pd2
0w = d0Pd0Pe0v = d0Pe20u =
e20rP
2i = e20i
3 = e20Pd0Pu = e2
0uP 2d0 = e0gP 3v = e0riP2e0 =
e0rjP2d0 = e0rPd0Pj = e0ikPj = e0iPd0z = e0j
2Pj =e0Pd2
0v = e0Pd0Pe0u = e0wP 3e0 = g2P 3u = griP 2d0 = gijP j =gPd2
0u = gPe0P2v = gvP 3e0 = gwP 3d0 = gzP 2j = gP 2e0Pv =
riPe20 = rjPd0Pe0 = rkPd2
0 = rlP 3e0 = rmP 3d0 = i2lP e0 =i2mPd0 = ijkPe0 = ijlPd0 = ik2Pd0 = j3Pe0 = j2kPd0 =jmP 2j = klP 2j = kmP 2i = kzP 2e0 = l2P 2i = lzP 2d0 = mPj2 =Pe0wP 2e0
(110) iP 3Q1 = x′P 3j = PQ1 P 2i
(001) h0P3x16,35 = Ph2P
2x16,32 = P 3h2x16,32
(101) e0x25,24
(111) d0P3R2 = Pd0P
2R2 = P 2d0PR2 = P 2e0R1 = R2 P 3d0
continued
148 ROBERT R. BRUNER
Stem 103 continued30 (10) h0iP
3Q1 = h0x′P 3j = h0Pjx18,20 = h0PQ1 P 2i = f0P
4x′ =Ph2ix18,20 = Ph2x
′P 2i = ix′P 3h2 = B2P4i = B4P
5h2
(01) h20P
3x16,35 = h0Ph2P2x16,32 = h0P
3h2x16,32
(11) h0d0P3R2 = h0e0x25,24 = h0Pd0P
2R2 = h0P2d0PR2 =
h0P2e0R1 = h0R2 P 3d0 = h2d0x25,24 = h2P
2d0R1 =d0P
2h2R1 = Ph2Pd0R1 = R1P4e0 = P 2D1P
4h2
31 (10) h20iP
3Q1 = h20x
′P 3j = h20Pjx18,20 = h2
0PQ1 P 2i =h0f0P
4x′ = h0Ph2ix18,20 = h0Ph2x′P 2i = h0ix
′P 3h2 =h0B2P
4i = h0B4P5h2 = h1e0P
4x′ = h1x′P 4e0 = h1P
2e0x18,20 =e0P
2h1x18,20 = e0x′P 4h1 = Ph1Pe0x18,20 = Ph1x
′P 3e0 =B1P
5e0 = Pe0x′P 3h1 = P 2h1x
′P 2e0 = x10,28P5h1
(01) h30P
3x16,35 = h20Ph2P
2x16,32 = h20P
3h2x16,32 = h21Ph1P
2x16,32 =h2
1P3h1x16,32 = h1Ph1P
2h1x16,32 = h1uP 3u = h1X1P5h1 =
h1QP 2u = h1PuP 2u = h1W1P4h1 = h3P
5Q1 = Ph31x16,32 =
Ph1uP 2u = Ph1X1P4h1 = Ph1P
3h1W1 = Ph1Q2 = Ph1QPu =
Ph1Pu2 = qP 4u = GP 6h1 = P 2h21W1 = P 2h1uQ = P 2h1uPu =
P 2h1X1P3h1 = u2P 3h1
(11) h20d0P
3R2 = h20e0x25,24 = h2
0Pd0P2R2 = h2
0P2d0PR2 =
h20P
2e0R1 = h20R2 P 3d0 = h0h2d0x25,24 = h0h2P
2d0R1 =h0d0P
2h2R1 = h0Ph2Pd0R1 = h0R1P4e0 = h0P
2D1P4h2 =
h1d0P4Q1 = h1Pd0P
3Q1 = h1Q1 P 4d0 = h1P2d0P
2Q1 =h1PQ1 P 3d0 = h2
2P4R2 = h2Ph2P
3R2 = h2P2h2P
2R2 =h2R1P
4d0 = h2P3h2PR2 = h2R2 P 4h2 = d0Ph1P
3Q1 =d0P
2h1P2Q1 = d0R1P
4h2 = d0Q1 P 4h1 = d0P3h1PQ1 =
Ph1Pd0P2Q1 = Ph1Q1 P 3d0 = Ph1P
2d0PQ1 = Ph22P
2R2 =Ph2P
2h2PR2 = Ph2R1P3d0 = Ph2P
3h2R2 = Pd0P2h1PQ1 =
Pd0R1P3h2 = Pd0Q1 P 3h1 = P 2h1Q1 P 2d0 = P 2h2
2R2 =P 2h2R1P
2d0 = x10,27P5h1
32 (1) d50P
2e0 = d40e0P
2d0 = d40Pd0Pe0 = d3
0e0Pd20 = d3
0gP 3e0 =d20e
20P
3e0 = d20e0gP 3d0 = d2
0e0Pe0P2e0 = d2
0gPd0P2e0 =
d20gPe0P
2d0 = d20Pe3
0 = d0e30P
3d0 = d0e20Pd0P
2e0 =d0e
20Pe0P
2d0 = d0e0gPd0P2d0 = d0e0Pd0Pe2
0 = d0g2P 4e0 =
d0gPd20Pe0 = d0iP
3v = d0jP3u = d0uP 3j = d0PjP 2u =
d0PuP 2j = d0PvP 2i = e30Pd0P
2d0 = e20gP 4e0 = e2
0Pd20Pe0 =
e0g2P 4d0 = e0gPd3
0 = e0gPe0P3e0 = e0gP 2e2
0 = e0iP3u =
e0PuP 2i = g2Pd0P3e0 = g2Pe0P
3d0 = g2P 2d0P2e0 =
gPe20P
2e0 = riP 3j = rPjP 2i = i3Pj = iPd0P2v = iPe0P
2u =iuP 3e0 = ivP 3d0 = izP 2i = iP 2d0Pv = iP 2e0Pu = jPd0P
2u =juP 3d0 = jP 2d0Pu = kP 4v = lP 4u = Pd0uP 2j = Pd0vP 2i =Pd0PjPu = Pe0uP 2i = uPjP 2d0 = wP 4j
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 149
Stem 103 continued35 (1) d3
0iP3d0 = d3
0Pd0P2i = d2
0e0P4j = d2
0gP 4i = d20iPd0P
2d0 =d20jP
4e0 = d20kP 4d0 = d2
0Pe0P3j = d2
0PjP 3e0 = d20P
2e0P2j =
d0e20P
4i = d0e0iP4e0 = d0e0jP
4d0 = d0e0Pd0P3j =
d0e0PjP 3d0 = d0e0P2d0P
2j = d0e0P2e0P
2i = d0giP 4d0 =d0gP 2d0P
2i = d0iPd30 = d0iPe0P
3e0 = d0iP2e2
0 = d0jPd0P3e0 =
d0jPe0P3d0 = d0jP
2d0P2e0 = d0kPd0P
3d0 = d0kP 2d20 =
d0lP5e0 = d0mP 5d0 = d0Pd0Pe0P
2j = d0Pd0PjP 2e0 =d0Pe2
0P2i = d0Pe0PjP 2d0 = e2
0iP4d0 = e2
0P2d0P
2i = e0gP 5j =e0iPd0P
3e0 = e0iPe0P3d0 = e0iP
2d0P2e0 = e0jPd0P
3d0 =e0jP
2d20 = e0kP 5e0 = e0lP
5d0 = e0Pd20P
2j = e0Pd0Pe0P2i =
e0Pd0PjP 2d0 = giPd0P3d0 = giP 2d2
0 = gjP 5e0 = gkP 5d0 =gPd2
0P2i = gPe0P
4j = gPjP 4e0 = gP 2e0P3j = gP 2jP 3e0 =
iPd0Pe0P2e0 = iPe2
0P2d0 = jPd2
0P2e0 = jPd0Pe0P
2d0 =kPd2
0P2d0 = kPe0P
4e0 = kP 2e0P3e0 = lPd0P
4e0 = lP e0P4d0 =
lP 2d0P3e0 = lP 2e0P
3d0 = mPd0P4d0 = mP 2d0P
3d0 =Pd2
0Pe0Pj
36 (1) h21P
4x18,20 = h1P2h1P
4x′ = h1x′P 6h1 = h1P
4h1x18,20 =Ph2
1P4x′ = Ph1x
′P 5h1 = Ph1P3h1x18,20 = B1P
7h1 =P 2h2
1x18,20 = P 2h1x′P 4h1 = x′P 3h2
1
150 ROBERT R. BRUNER
Stem 1044 (1) h6c2
5 (1) h0h6c2
6 (1) h20h6c2 = h1h6f1 = h2h6e1
8 (1) h3x7,81
9 (10) x9,97
(01) h0h3x7,81 = h2x8,93
10 (1) h0x9,97
11 (10) h6z
(01) h20x9,97
(11) h3x10,76
12 (100) x12,80
(010) h0h6z = h21h6u = h2x11,80 = h6f0i = h6Ph1q = h6Ph2r
(001) h30x9,97
(011) h0h3x10,76
13 (10) h0x12,80 = h1x12,78
(01) h40x9,97 = h2
0h3x10,76 = g2B4
14 (1) h50x9,97 = h3
0h3x10,76 = h0g2B4 = h23x12,55 = xx9,40 = rx8,51
16 (1) d0x12,55
17 (10) h6P3e0
(01) h0d0x12,55 = h2Px12,60 = Ph2x12,60 = B2Q1
18 (100) gx14,46
(010) x18,68
(001) h20d0x12,55 = h0h2Px12,60 = h0h6P
3e0 = h0Ph2x12,60 =h0B2Q1 = h2h6P
3d0 = h2Ph2x12,55 = h2B2x′ = h5PR2 =
h6d0P3h2 = h6Ph2P
2d0 = h6Pd0P2h2
19 (10) d0rB4 = e0x15,47 = gx15,43 = uB23 = vx10,27 = vx10,28 = wB21
(01) h0x18,68 = h2x18,63
20 (1) h20x18,68 = h0h2x18,63
21 (10) e0x17,50
(01) h30x18,68 = h2
0h2x18,63 = h31x18,63 = h1Ph1x15,56
(11) d0x17,52
22 (10) d0g3r = d0gm2 = d0r
3 = d0w2 = e2
0g2r = e2
0m2 = e0glm =
e0vw = g2km = g2l2 = guw = gv2 = rkw = rlv = rmu
(01) d30x10,27 = d3
0x10,28 = d20e0B21 = d2
0gQ1 = d0e20Q1 = d0e0gx′ =
d0Pe0B23 = e30x
′ = e0Pd0B23 = e0Pe0x10,27 = e0Pe0x10,28 =g2PQ1 = gPd0x10,27 = gPd0x10,28 = gPe0B21 = uR2
24 (1) d20x16,32 = e0Px16,35 = Pd0x16,38 = Pe0x16,35
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 151
Stem 104 continued25 (10) d3
0e0w = d30gv = d3
0rl = d20e
20v = d2
0e0gu = d20e0rk = d2
0grj =d20mz = d0e
30u = d0e
20rj = d0e0gri = d0e0lz = d0g
2Pv = d0gkz =d0gPe0w = d0rmPe0 = d0ilm = d0jkm = d0jl
2 = d0k2l = e3
0ri =e20gPv = e2
0kz = e20Pe0w = e0g
2Pu = e0gjz = e0gPd0w =e0gPe0v = e0rlPe0 = e0rmPd0 = e0ikm = e0il
2 = e0j2m =
e0jkl = e0k3 = g2rPj = g2iz = g2Pd0v = g2Pe0u = grkPe0 =
grlPd0 = gijm = gikl = gj2l = gjk2 = iuv = ju2 = m2Pj
(01) d20ix
′ = d0e0PR2 = d0gR1 = d0jPQ1 = d0Pd0P2D1 =
d0Pe0R2 = d0B4P2d0 = d0Q1 Pj = e2
0R1 = e0iPQ1 =e0Pd0R2 = e0x
′Pj = iPd0B21 = iPe0Q1 = jPd0Q1 = jPe0x′ =
kPd0x′ = lP 2Q1 = Pd2
0B4 = x10,27P2j = x10,28P
2j = B23P2i =
P 2e0x13,35
(11) mx18,20
26 (10) h1P2x17,50 = Ph1x21,43 = Ph1Px17,50 = P 2h1x17,50
(01) h0d20ix
′ = h0d0e0PR2 = h0d0gR1 = h0d0jPQ1 =h0d0Pd0P
2D1 = h0d0Pe0R2 = h0d0B4P2d0 = h0d0Q1 Pj =
h0e20R1 = h0e0iPQ1 = h0e0Pd0R2 = h0e0x
′Pj = h0iPd0B21 =h0iPe0Q1 = h0jPd0Q1 = h0jPe0x
′ = h0kPd0x′ = h0lP
2Q1 =h0mx18,20 = h0Pd2
0B4 = h0x10,27P2j = h0x10,28P
2j =h0B23P
2i = h0P2e0x13,35 = h2d
20PR2 = h2d0e0R1 =
h2d0iPQ1 = h2d0Pd0R2 = h2d0x′Pj = h2gP 2R2 =
h2iPd0Q1 = h2iPe0x′ = h2jPd0x
′ = h2kP 2Q1 = h2lx18,20 =h2B4P
3e0 = h2B21P2j = h2x10,27P
2i = h2x10,28P2i =
h2P2d0x13,35 = h2P
2e0P2D1 = h4P
3R2 = d20Ph2R2 =
d20Pd0R1 = d0f0P
2Q1 = d0Ph2iQ1 = d0Ph2jx′ =
d0B2P2j = d0P
2h2x13,35 = e0f0x18,20 = e0Ph2ix′ = e0B2P
2i =e0P
2h2P2D1 = e0B4P
3h2 = e0R1P2e0 = f0Pd0PQ1 =
f0x′P 2e0 = f0Q1 P 2d0 = gPh2PR2 = gP 2h2R2 = gR1P
2d0 =Ph2kPQ1 = Ph2Pd0x13,35 = Ph2Pe0P
2D1 = Ph2B4P2e0 =
Ph2B21Pj = D2P4d0 = iB2P
2e0 = iP 2h2x10,27 = iP 2h2x10,28 =jB2P
2d0 = jP 2h2B21 = kP 2h2Q1 = lP 2h2x′ = B2Pd0Pj =
Pe20R1 = Pe0P
2h2B4 = B5 P 3d0 = PD2P3d0
continued
152 ROBERT R. BRUNER
Stem 104 continued27 (1) h2
0d20ix
′ = h20d0e0PR2 = h2
0d0gR1 = h20d0jPQ1 =
h20d0Pd0P
2D1 = h20d0Pe0R2 = h2
0d0B4P2d0 = h2
0d0Q1 Pj =h2
0e20R1 = h2
0e0iPQ1 = h20e0Pd0R2 = h2
0e0x′Pj = h2
0iPd0B21 =h2
0iPe0Q1 = h20jPd0Q1 = h2
0jPe0x′ = h2
0kPd0x′ = h2
0lP2Q1 =
h20mx18,20 = h2
0Pd20B4 = h2
0x10,27P2j = h2
0x10,28P2j =
h20B23P
2i = h20P
2e0x13,35 = h0h2d20PR2 = h0h2d0e0R1 =
h0h2d0iPQ1 = h0h2d0Pd0R2 = h0h2d0x′Pj = h0h2gP 2R2 =
h0h2iPd0Q1 = h0h2iPe0x′ = h0h2jPd0x
′ = h0h2kP 2Q1 =h0h2lx18,20 = h0h2B4P
3e0 = h0h2B21P2j = h0h2x10,27P
2i =h0h2x10,28P
2i = h0h2P2d0x13,35 = h0h2P
2e0P2D1 =
h0h4P3R2 = h0d
20Ph2R2 = h0d
20Pd0R1 = h0d0f0P
2Q1 =h0d0Ph2iQ1 = h0d0Ph2jx
′ = h0d0B2P2j = h0d0P
2h2x13,35 =h0e0f0x18,20 = h0e0Ph2ix
′ = h0e0B2P2i = h0e0P
2h2P2D1 =
h0e0B4P3h2 = h0e0R1P
2e0 = h0f0Pd0PQ1 = h0f0x′P 2e0 =
h0f0Q1 P 2d0 = h0gPh2PR2 = h0gP 2h2R2 = h0gR1P2d0 =
h0Ph2kPQ1 = h0Ph2Pd0x13,35 = h0Ph2Pe0P2D1 =
h0Ph2B4P2e0 = h0Ph2B21Pj = h0D2P
4d0 = h0iB2P2e0 =
h0iP2h2x10,27 = h0iP
2h2x10,28 = h0jB2P2d0 = h0jP
2h2B21 =h0kP 2h2Q1 = h0lP
2h2x′ = h0B2Pd0Pj = h0Pe2
0R1 =h0Pe0P
2h2B4 = h0B5 P 3d0 = h0PD2P3d0 = h1d
20Pd0x
′ =h1d0e0P
2Q1 = h1d0gx18,20 = h1d0Pe0PQ1 = h1d0Q1 P 2e0 =h1d0B21P
2d0 = h1e20x18,20 = h1e0Pd0PQ1 = h1e0x
′P 2e0 =h1e0Q1 P 2d0 = h1gx′P 2d0 = h1Pd2
0B21 = h1Pd0Pe0Q1 =h1Pe2
0x′ = h1x10,27P
3e0 = h1x10,28P3e0 = h1B23P
3d0 =h2
2d20R1 = h2
2e0P2R2 = h2
2iPd0x′ = h2
2jP2Q1 = h2
2kx18,20 =h2
2Pe0PR2 = h22B4P
3d0 = h22Q1 P 2j = h2
2B21P2i = h2
2PjPQ1 =h2
2P2d0P
2D1 = h22P
2e0R2 = h2h4x25,24 = h2d0f0x18,20 =h2d0Ph2ix
′ = h2d0B2P2i = h2d0P
2h2P2D1 = h2d0B4P
3h2 =h2d0R1P
2e0 = h2e0Ph2PR2 = h2e0P2h2R2 = h2e0R1P
2d0 =h2f0x
′P 2d0 = h2gPh2R1 = h2Ph2jPQ1 = h2Ph2Pd0P2D1 =
h2Ph2Pe0R2 = h2Ph2B4P2d0 = h2Ph2Q1 Pj = h2iB2P
2d0 =h2iP
2h2B21 = h2jP2h2Q1 = h2kP 2h2x
′ = h2Pd0Pe0R1 =h2Pd0P
2h2B4 = h4P2h2R1 = c0iPR2 = c0jR1 =
d30Ph1x
′ = d20B1P
2d0 = d20P
2h1B21 = d0e0Ph1PQ1 =d0e0P
2h1Q1 = d0e0P2h2R1 = d0f0P
2h2x′ = d0gP 2h1x
′ =d0Ph1Pd0B21 = d0Ph1Pe0Q1 = d0Ph2
2P2D1 = d0Ph2Pe0R1 =
d0Ph2P2h2B4 = d0iB2P
2h2 = d0B1Pd20 = d0B23P
3h1 =e20P
2h1x′ = e0Ph1Pd0Q1 = e0Ph1Pe0x
′ = e0Ph22R2 =
e0Ph2Pd0R1 = e0B1P3e0 = e0x10,27P
3h1 = e0x10,28P3h1 =
f0Ph2Pd0x′ = f0B2P
3d0 = f0B21P3h2 = gPh1Pd0x
′ =gB1P
3d0 = gB21P3h1 = Ph1x10,27P
2e0 = Ph1x10,28P2e0 =
Ph1B23P2d0 = Ph2
2iB21 = Ph22jQ1 = Ph2
2kx′ = Ph22Pd0B4 =
Ph2iB2Pd0 = AP 5h2 = Pc0iR2 = kB2P3h2 = B1Pe0P
2e0 =Pd0P
2h1B23 = Pe0P2h1x10,27 = Pe0P
2h1x10,28 = PAP 4h2 =P 2c0x16,37
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 153
Stem 104 continued28 (10) iP 2R2 = PjR1 = R2 P 2i
(01) d60g = d5
0e20 = d3
0e0gPe0 = d30g
2Pd0 = d30iu = d2
0e30Pe0 =
d20e
20gPd0 = d2
0ri2 = d2
0jPv = d20kPu = d2
0vPj = d0e40Pd0 =
d0e0g2P 2e0 = d0e0iPv = d0e0jPu = d0e0uPj = d0g
3P 2d0 =d0g
2Pe20 = d0giPu = d0r
2P 2d0 = d0rjPj = d0ijz = d0iPd0w =d0iPe0v = d0jPd0v = d0jPe0u = d0kPd0u = d0lP
2v =d0mP 2u = e3
0gP 2e0 = e20g
2P 2d0 = e20gPe2
0 = e20iPu =
e0g2Pd0Pe0 = e0riP j = e0i
2z = e0iPd0v = e0iPe0u =e0jPd0u = e0kP 2v = e0lP
2u = e0wP 2j = g3Pd20 = giPd0u =
gjP 2v = gkP 2u = gvP 2j = gwP 2i = gPjPv = r2Pd20 =
rijPe0 = rikPd0 = rj2Pd0 = rlP 2j = rmP 2i = rzP 2e0 =i3m = i2jl = i2k2 = ij2k = j4 = jwP 2e0 = kPe0Pv = kvP 2e0 =kwP 2d0 = kzPj = lPd0Pv = lP e0Pu = luP 2e0 = lvP 2d0 =mPd0Pu = muP 2d0 = Pd0z
2 = Pe0wPj
29 (1) h0iP2R2 = h0PjR1 = h0R2 P 2i = h1P
3x16,35 = f0x25,24 =Ph1P
2x16,35 = Ph2iR1 = P 2h1Px16,35 = R1P3j = P 3h1x16,35 =
P 3h2x16,33
31 (1) d50Pj = d4
0iPe0 = d40jPd0 = d3
0e0iPd0 = d30gP 2j = d3
0kP 2e0 =d30lP
2d0 = d20e
20P
2j = d20e0gP 2i = d2
0e0jP2e0 = d2
0e0kP 2d0 =d20e0Pe0Pj = d2
0giP 2e0 = d20gjP 2d0 = d2
0gPd0Pj = d20jPe2
0 =d20kPd0Pe0 = d2
0lPd20 = d2
0mP 3e0 = d0e30P
2i = d0e20iP
2e0 =d0e
20jP
2d0 = d0e20Pd0Pj = d0e0giP 2d0 = d0e0iPe2
0 =d0e0jPd0Pe0 = d0e0kPd2
0 = d0e0lP3e0 = d0e0mP 3d0 =
d0g2P 3j = d0giPd0Pe0 = d0gjPd2
0 = d0gkP 3e0 = d0glP 3d0 =d0lP e0P
2e0 = d0mPd0P2e0 = d0mPe0P
2d0 = e30iP
2d0 =e20gP 3j = e2
0iPd0Pe0 = e20jPd2
0 = e20kP 3e0 = e2
0lP3d0 =
e0giPd20 = e0gjP 3e0 = e0gkP 3d0 = e0gPe0P
2j = e0gPjP 2e0 =e0kPe0P
2e0 = e0lPd0P2e0 = e0lP e0P
2d0 = e0mPd0P2d0 =
g2iP 3e0 = g2jP 3d0 = g2Pd0P2j = g2Pe0P
2i = g2PjP 2d0 =gjPe0P
2e0 = gkPd0P2e0 = gkPe0P
2d0 = glPd0P2d0 =
gmP 4e0 = gPe20Pj = rP 4v = i2P 2v = ijP 2u = iuP 2j = ivP 2i =
iP jPu = juP 2i = kPe30 = lPd0Pe2
0 = mPd20Pe0 = zP 3u
154 ROBERT R. BRUNER
Stem 104 continued34 (01) d3
0iP2i = d2
0rP4d0 = d2
0i2P 2d0 = d2
0jP3j = d2
0PjP 2j =d0e0iP
3j = d0e0PjP 2i = d0rPd0P3d0 = d0rP
2d20 = d0i
2Pd20 =
d0ijP3e0 = d0ikP 3d0 = d0iPe0P
2j = d0iP jP 2e0 = d0j2P 3d0 =
d0jPd0P2j = d0jPe0P
2i = d0jP jP 2d0 = d0kPd0P2i =
d0lP4j = d0mP 4i = d0Pd0Pj2 = d0zP 4e0 = e0rP
5e0 =e0i
2P 3e0 = e0ijP3d0 = e0iPd0P
2j = e0iPe0P2i = e0iP jP 2d0 =
e0jPd0P2i = e0kP 4j = e0lP
4i = e0zP 4d0 = grP 5d0 =gi2P 3d0 = giPd0P
2i = gjP 4j = gkP 4i = gPjP 3j =gP 2j2 = rPd2
0P2d0 = rPe0P
4e0 = rP 2e0P3e0 = i2Pe0P
2e0 =ijPd0P
2e0 = ijPe0P2d0 = ikPd0P
2d0 = ilP 4e0 = imP 4d0 =iPd0Pe0Pj = j2Pd0P
2d0 = jkP 4e0 = jlP 4d0 = jPd20Pj =
k2P 4d0 = kPe0P3j = kPjP 3e0 = kP 2e0P
2j = lPd0P3j =
lP jP 3d0 = lP 2d0P2j = lP 2e0P
2i = mP 2d0P2i = Pd0zP 3e0 =
Pe0zP 3d0 = zP 2d0P2e0
(11) P 6Q1
35 (1) h0P6Q1 = h2P
4x18,20 = P 2h2P4x′ = x′P 6h2 = P 4h2x18,20
36 (1) h20P
6Q1 = h0h2P4x18,20 = h0P
2h2P4x′ = h0x
′P 6h2 =h0P
4h2x18,20
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 155
Stem 1057 (1) x7,92
8 (1) h0x7,92 = g2D3
9 (1) x9,99
10 (100) h23x8,75 = g2A
(001) h0x9,99 = h1x9,97 = h6v
(111) g2A′
11 (10) h0h23x8,75 = h0g2A = h2g2D2 = h3c2Q2 = h3f1D2 = h4x10,63 =
n1y
(11) h0g2A′
13 (10) h23x11,61 = g2X1 = xG21
(01) h1x12,80 = h4x12,55 = c0x10,76
(11) h6d30 = h6e0Pe0 = h6gPd0
14 (1) h0h23x11,61 = h0h6d
30 = h0h6e0Pe0 = h0h6gPd0 = h0g2X1 =
h0xG21 = h2h6d0Pe0 = h2h6e0Pd0 = h6d0e0Ph2 = B2Q2
15 (10) x15,68
(01) d0x11,61
16 (100) h6P2j
(010) h0x15,68
(001) h0d0x11,61 = h2x15,65
17 (100) h0h6P2j = h2h6P
2i = h6iP2h2
(010) h20x15,68
(001) h20d0x11,61 = h0h2x15,65 = h1d0x12,55 = h2Ph2x11,61 =
Ph1x12,64 = B1B21
18 (10) g2B23 = wB4
(01) h20h6P
2j = h0h2h6P2i = h0h6iP
2h2 = h1h6P3e0 = h4x17,52 =
h6e0P3h1 = h6Ph1P
2e0 = h6Pe0P2h1
19 (1) h1x18,68
20 (1) e0x16,48 = gx16,42
21 (10) g3w = g2rm = r2w = m3
(01) d20e0B4 = d0gx13,35 = d0jB23 = d0kx10,27 = d0kx10,28 = d0lB21 =
d0mQ1 = e20x13,35 = e0gP 2D1 = e0iB23 = e0jx10,27 = e0jx10,28 =
e0kB21 = e0lQ1 = e0mx′ = g2R2 = gix10,27 = gix10,28 =gjB21 = gkQ1 = glx′ = gPe0B4
23 (1) ix16,38 = jx16,35 = kx16,32 = Pd0x15,41
continued
156 ROBERT R. BRUNER
Stem 105 continued24 (01) d2
0x16,33 = d0iP2D1 = d0jR2 = e0iR2 = rPd0x
′ = i2B21 =ijQ1 = ikx′ = iPd0B4 = j2x′ = lPR2 = mR1 = Pe0x16,37 =zPQ1 = Pjx13,35
(11) d20e0g
3 = d20e0r
2 = d20lw = d2
0mv = d0e30g
2 = d0e0kw = d0e0lv =d0e0mu = d0grz = d0gjw = d0gkv = d0glu = d0rjm = d0rkl =e50g = e2
0rz = e20jw = e2
0kv = e20lu = e0giw = e0gjv = e0gku =
e0rim = e0rjl = e0rk2 = g4Pe0 = g2iv = g2ju = gr2Pe0 = gril =
grjk = gmPv = kmz = l2z = mPe0w
26 (1) P 2x18,50
27 (10) d50m = d4
0e0l = d40gk = d3
0e20k = d3
0e0gj = d30g
2i = d20e
30j =
d20e
20gi = d2
0e0mPe0 = d20glPe0 = d2
0gmPd0 = d20rPu = d0e
40i =
d0e20lP e0 = d0e
20mPd0 = d0e0g
2Pj = d0e0gkPe0 = d0e0glPd0 =d0g
2jPe0 = d0g2kPd0 = d0rPd0u = d0i
2w = d0ijv = d0iku =d0j
2u = d0zPv = e30gPj = e3
0kPe0 = e30lPd0 = e2
0gjPe0 =e20gkPd0 = e0g
2iPe0 = e0g2jPd0 = e0gmP 2e0 = e0rP
2v =e0i
2v = e0iju = e0zPu = g3iPd0 = g2lP 2e0 = g2mP 2d0 =grP 2u = gi2u = gmPe2
0 = r2iPd0 = ri2k = rij2 = rPe0Pv =rvP 2e0 = rwP 2d0 = rzPj = ilPv = imPu = iz2 = jkPv =jlPu = jwPj = k2Pu = kvPj = luPj = Pd0vz = Pe0uz
(01) h0P2x18,50 = h2
1P2x17,50 = h1Ph1x21,43 = h1Ph1Px17,50 =
h1P2h1x17,50 = Ph2
1x17,50 = Ph2x22,39 = P 3h1x14,42
30 (10) d0x′P 3d0 = e0P
4Q1 = gP 4x′ = Pd0x′P 2d0 = Pe0P
3Q1 =Q1 P 4e0 = B21P
4d0 = P 2e0P2Q1 = PQ1 P 3e0
(01) d40ij = d3
0e0i2 = d3
0rP2e0 = d3
0kPj = d30Pd0z = d2
0e0rP2d0 =
d20e0jP j = d2
0giP j = d20rPd0Pe0 = d2
0ikPe0 = d20ilPd0 =
d20j
2Pe0 = d20jkPd0 = d2
0mP 2j = d0e20iP j = d0e0rPd2
0 =d0e0ijPe0 = d0e0ikPd0 = d0e0j
2Pd0 = d0e0lP2j = d0e0mP 2i =
d0e0zP 2e0 = d0grP 3e0 = d0gi2Pe0 = d0gijPd0 = d0gkP 2j =d0glP 2i = d0gzP 2d0 = d0imP 2e0 = d0jlP
2e0 = d0jmP 2d0 =d0k
2P 2e0 = d0klP 2d0 = d0lP e0Pj = d0mPd0Pj = d0Pe20z =
e20rP
3e0 = e20i
2Pe0 = e20ijPd0 = e2
0kP 2j = e20lP
2i = e20zP 2d0 =
e0grP 3d0 = e0gi2Pd0 = e0gjP 2j = e0gkP 2i = e0gPj2 =e0rPe0P
2e0 = e0ilP2e0 = e0imP 2d0 = e0jkP 2e0 = e0jlP
2d0 =e0k
2P 2d0 = e0kPe0Pj = e0lPd0Pj = e0Pd0Pe0z = g2iP 2j =g2jP 2i = grPd0P
2e0 = grPe0P2d0 = gikP 2e0 = gilP 2d0 =
gj2P 2e0 = gjkP 2d0 = gjPe0Pj = gkPd0Pj = gmP 3j = gPd20z =
rPe30 = ilPe2
0 = imPd0Pe0 = jkPe20 = jlPd0Pe0 = jmPd2
0 =k2Pd0Pe0 = klPd2
0 = kmP 3e0 = l2P 3e0 = lmP 3d0 = uP 3v =vP 3u = PuP 2v = PvP 2u
(11) d0Pd0x18,20
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 157
Stem 105 continued31 (1) h0d0Pd0x18,20 = h0d0x
′P 3d0 = h0e0P4Q1 = h0gP 4x′ =
h0Pd0x′P 2d0 = h0Pe0P
3Q1 = h0Q1 P 4e0 = h0B21P4d0 =
h0P2e0P
2Q1 = h0PQ1 P 3e0 = h2d0P4Q1 = h2e0P
4x′ =h2Pd0P
3Q1 = h2x′P 4e0 = h2Q1 P 4d0 = h2P
2d0P2Q1 =
h2P2e0x18,20 = h2PQ1 P 3d0 = d0Ph2P
3Q1 = d0P2h2P
2Q1 =d0Q1 P 4h2 = d0P
3h2PQ1 = e0P2h2x18,20 = e0x
′P 4h2 =Ph2Pd0P
2Q1 = Ph2Pe0x18,20 = Ph2x′P 3e0 = Ph2Q1 P 3d0 =
Ph2P2d0PQ1 = B2P
5e0 = Pd0P2h2PQ1 = Pd0Q1 P 3h2 =
Pe0x′P 3h2 = P 2h2x
′P 2e0 = P 2h2Q1 P 2d0 = x10,27P5h2 =
x10,28P5h2
32 (1) h20d0Pd0x18,20 = h2
0d0x′P 3d0 = h2
0e0P4Q1 = h2
0gP 4x′ =h2
0Pd0x′P 2d0 = h2
0Pe0P3Q1 = h2
0Q1 P 4e0 = h20B21P
4d0 =h2
0P2e0P
2Q1 = h20PQ1 P 3e0 = h0h2d0P
4Q1 =h0h2e0P
4x′ = h0h2Pd0P3Q1 = h0h2x
′P 4e0 = h0h2Q1 P 4d0 =h0h2P
2d0P2Q1 = h0h2P
2e0x18,20 = h0h2PQ1 P 3d0 =h0d0Ph2P
3Q1 = h0d0P2h2P
2Q1 = h0d0Q1 P 4h2 =h0d0P
3h2PQ1 = h0e0P2h2x18,20 = h0e0x
′P 4h2 =h0Ph2Pd0P
2Q1 = h0Ph2Pe0x18,20 = h0Ph2x′P 3e0 =
h0Ph2Q1 P 3d0 = h0Ph2P2d0PQ1 = h0B2P
5e0 =h0Pd0P
2h2PQ1 = h0Pd0Q1 P 3h2 = h0Pe0x′P 3h2 =
h0P2h2x
′P 2e0 = h0P2h2Q1 P 2d0 = h0x10,27P
5h2 =h0x10,28P
5h2 = h22d0P
4x′ = h22x
′P 4d0 = h22P
2d0x18,20 =h2d0P
2h2x18,20 = h2d0x′P 4h2 = h2Ph2Pd0x18,20 =
h2Ph2x′P 3d0 = h2B2P
5d0 = h2Pd0x′P 3h2 = h2P
2h2x′P 2d0 =
h2B21P5h2 = c0P
4R2 = d0Ph22x18,20 = d0Ph2x
′P 3h2 =d0B2P
5h2 = d0P2h2
2x′ = Ph2
2x′P 2d0 = Ph2B2P
4d0 =Ph2Pd0P
2h2x′ = Ph2B21P
4h2 = Pc0P3R2 = B2Pd0P
4h2 =B2P
2h2P3d0 = B2P
2d0P3h2 = P 2h2B21P
3h2 = P 2c0P2R2 =
R2 P 4c0 = P 3c0PR2
continued
158 ROBERT R. BRUNER
Stem 105 continued33 (10) P 5R2
(01) d30P
3u = d20Pd0P
2u = d20uP 3d0 = d2
0P2d0Pu = d0e0P
4v =d0gP 4u = d0riP
3d0 = d0rPd0P2i = d0i
3Pd0 = d0ijP2j =
d0ikP 2i = d0iP j2 = d0j2P 2i = d0Pd2
0Pu = d0Pd0uP 2d0 =d0Pe0P
3v = d0vP 4e0 = d0wP 4d0 = d0zP 3j = d0P2e0P
2v =d0PvP 3e0 = e2
0P4u = e0rP
4j = e0i2P 2j = e0ijP
2i =e0Pd0P
3v = e0Pe0P3u = e0uP 4e0 = e0vP 4d0 = e0P
2d0P2v =
e0P2e0P
2u = e0PuP 3e0 = e0PvP 3d0 = grP 4i = gi2P 2i =gPd0P
3u = guP 4d0 = gP 2d0P2u = gPuP 3d0 = riPd0P
2d0 =rjP 4e0 = rkP 4d0 = rPe0P
3j = rPjP 3e0 = rP 2e0P2j =
i2jP 2e0 = i2kP 2d0 = i2Pe0Pj = ij2P 2d0 = ijPd0Pj =ilP 3j = izP 3e0 = jkP 3j = jzP 3d0 = kPjP 2j = lP jP 2i =Pd3
0u = Pd0Pe0P2v = Pd0vP 3e0 = Pd0wP 3d0 = Pd0zP 2j =
Pd0P2e0Pv = Pe2
0P2u = Pe0uP 3e0 = Pe0vP 3d0 = Pe0zP 2i =
Pe0P2d0Pv = Pe0P
2e0Pu = uP 2e20 = vP 2d0P
2e0 = wP 2d20 =
zPjP 2d0
34 (1) h0P5R2 = h2P
4R1 = P 2h2x25,24 = P 4h2R1
35 (1) h20P
5R2 = h0h2P4R1 = h0P
2h2x25,24 = h0P4h2R1 =
h1P6Q1 = Ph1P
5Q1 = P 2h1P4Q1 = R1P
6h2 = Q1 P 6h1 =P 3h1P
3Q1 = PQ1 P 5h1 = P 4h1P2Q1
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 159
Stem 1066 (1) C1
8 (1) h3x7,84
9 (10) g2H1
(01) h0h3x7,84 = h4x8,75
10 (01) h1x9,99
(11) h3x9,86
12 (10) h4x11,61
(01) h6d0k = h6e0j = h6gi
(11) h23x10,65 = e1G21 = g2x8,32
13 (1) h0h4x11,61 = h0h6d0k = h0h6e0j = h0h6gi = h2h6d0j =h2h6e0i = h2x12,78 = h6f0Pe0 = h6Ph2l = D2B2
14 (10) x14,79
(01) h20h4x11,61 = h2
0h6d0k = h20h6e0j = h2
0h6gi = h0h2h6d0j =h0h2h6e0i = h0h2x12,78 = h0h6f0Pe0 = h0h6Ph2l = h0D2B2 =h2
1x12,80 = h1h4x12,55 = h1h6d30 = h1h6e0Pe0 = h1h6gPd0 =
h1c0x10,76 = h22h6d0i = h2h6f0Pd0 = h2h6Ph2k = h6d0f0Ph2 =
h6d0gPh1 = h6e20Ph1 = Q3u = B1B3
15 (100) g2x7,40 = rx9,51 = A′w = Aw
(010) Ph1x10,76
(001) h0x14,79
16 (1) h20x14,79 = h1x15,68
20 (100) g3N = g2nm = g2rt = nrw = r2N = tm2
(001) d0rx10,27 = d0rx10,28 = d0lB4 = e0rB21 = e0kB4 = grQ1 =gjB4 = mx13,35 = zB23
(011) x′2
21 (1) h0x′2
22 (10) rx16,32 = ix15,41
(01) h20x
′2
(11) d0x18,55 = e0x18,50
23 (100) d0e0g2m = d0e0rw = d0g
3l = d0grv = d0r2l = e3
0gm = e20g
2l =e20rv = e0g
3k = e0gru = e0r2k = g4j = gr2j = gwz = rmz =
jmw = klw = kmv = l2v = lmu
(010) d0ux′ = jx16,37 = kx16,33 = Pd0x15,43 = vPQ1 = zR2 =Q1 Pv = B21Pu
(001) h30x
′2 = h0rx16,32 = h0ix15,41 = xx18,20
(011) rix′ = i2B4
24 (1) h40x
′2 = h20rx16,32 = h2
0ix15,41 = h0xx18,20 = h0rix′ = h0i
2B4 =h3ix16,32 = x9,40P
2i
continued
160 ROBERT R. BRUNER
Stem 106 continued25 (10) P 2x17,52
(01) h50x
′2 = h30rx16,32 = h3
0ix15,41 = h20xx18,20 = h2
0rix′ = h2
0i2B4 =
h0h3ix16,32 = h0x9,40P2i = h3rx18,20 = h3i
2x′ = h3B4P2i
26 (100) d40gr = d3
0e20r = d3
0km = d30l
2 = d20e0gz = d2
0e0jm = d20e0kl =
d20gim = d2
0gjl = d20gk2 = d2
0u2 = d0e
30z = d0e
20im = d0e
20jl =
d0e20k
2 = d0e0grPe0 = d0e0gil = d0e0gjk = d0g2rPd0 = d0g
2ik =d0g
2j2 = d0riu = d0lmPe0 = d0m2Pd0 = d0vPv = d0wPu =
e30rPe0 = e3
0il = e30jk = e2
0grPd0 = e20gik = e2
0gj2 = e0g2ij =
e0gmPj = e0kmPe0 = e0l2Pe0 = e0lmPd0 = e0uPv = e0vPu =
g3i2 = g2lP j = g2Pe0z = gjmPe0 = gklPe0 = gkmPd0 =gl2Pd0 = guPu = rjPv = rkPu = rvPj = ivz = juz = Pd0uw =Pd0v
2 = Pe0uv
(001) h60x
′2 = h40rx16,32 = h4
0ix15,41 = h30xx18,20 = h3
0rix′ =
h30i
2B4 = h20h3ix16,32 = h2
0x9,40P2i = h0h3rx18,20 = h0h3i
2x′ =h0h3B4P
2i = h23P
2x16,32 = g2iP2i = Ph2x21,43 = xrP 2i = xi3 =
x11,35P2i
(101) r2i2
(011) d30PQ1 = d2
0Pd0Q1 = d20Pe0x
′ = d0e0Pd0x′ = d0gP 2Q1 =
d0B21P2e0 = d0x10,27P
2d0 = d0x10,28P2d0 = e2
0P2Q1 =
e0Pe0PQ1 = e0Q1 P 2e0 = e0B21P2d0 = gPd0PQ1 = gx′P 2e0 =
gQ1 P 2d0 = Pd20x10,27 = Pd2
0x10,28 = Pd0Pe0B21 = Pe20Q1 =
B23P3e0
(111) e0gx18,20
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 161
Stem 106 continued
27 (1) h70x
′2 = h50rx16,32 = h5
0ix15,41 = h40xx18,20 = h4
0rix′ =
h40i
2B4 = h30h3ix16,32 = h3
0x9,40P2i = h2
0h3rx18,20 = h20h3i
2x′ =h2
0h3B4P2i = h0h
23P
2x16,32 = h0d30PQ1 = h0d
20Pd0Q1 =
h0d20Pe0x
′ = h0d0e0Pd0x′ = h0d0gP 2Q1 = h0d0B21P
2e0 =h0d0x10,27P
2d0 = h0d0x10,28P2d0 = h0e
20P
2Q1 = h0e0gx18,20 =h0e0Pe0PQ1 = h0e0Q1 P 2e0 = h0e0B21P
2d0 = h0gPd0PQ1 =h0gx′P 2e0 = h0gQ1 P 2d0 = h0g2iP
2i = h0Ph2x21,43 =h0xrP 2i = h0xi3 = h0r
2i2 = h0Pd20x10,27 = h0Pd2
0x10,28 =h0Pd0Pe0B21 = h0Pe2
0Q1 = h0B23P3e0 = h0x11,35P
2i =h1P
2x18,50 = h2d20Pd0x
′ = h2d0e0P2Q1 = h2d0gx18,20 =
h2d0Pe0PQ1 = h2d0Q1 P 2e0 = h2d0B21P2d0 = h2e
20x18,20 =
h2e0Pd0PQ1 = h2e0x′P 2e0 = h2e0Q1 P 2d0 = h2gx′P 2d0 =
h2Pd20B21 = h2Pd0Pe0Q1 = h2Pe2
0x′ = h2x10,27P
3e0 =h2x10,28P
3e0 = h2B23P3d0 = h2
3ix18,20 = h23x
′P 2i =h4Pd0x18,20 = h4x
′P 3d0 = c0Pd0x16,32 = d30Ph2x
′ = d20B2P
2d0 =d20P
2h2B21 = d0e0Ph2PQ1 = d0e0P2h2Q1 = d0gP 2h2x
′ =d0Ph2Pd0B21 = d0Ph2Pe0Q1 = d0Pc0x16,32 = d0B2Pd2
0 =d0B23P
3h2 = e20P
2h2x′ = e0Ph2Pd0Q1 = e0Ph2Pe0x
′ =e0B2P
3e0 = e0x10,27P3h2 = e0x10,28P
3h2 = gPh2Pd0x′ =
gB2P3d0 = gB21P
3h2 = Ph1Px18,50 = Ph2x10,27P2e0 =
Ph2x10,28P2e0 = Ph2B23P
2d0 = B2Pe0P2e0 = Q2P
4e0 =B3P
4d0 = Pd0P2h2B23 = Pe0P
2h2x10,27 = Pe0P2h2x10,28 =
P 2h1x18,50 = P 2c0x16,38
28 (1) d0P2x16,32 = P 2d0x16,32
29 (100) d0ix18,20
(010) d40Pv = d3
0e0Pu = d30rPj = d3
0iz = d30Pd0v = d3
0Pe0u =d20e0Pd0u = d2
0gP 2v = d20riPe0 = d2
0rjPd0 = d20i
2l = d20ijk =
d20j
3 = d20wP 2e0 = d0e
20P
2v = d0e0gP 2u = d0e0riPd0 =d0e0i
2k = d0e0ij2 = d0e0Pe0Pv = d0e0vP 2e0 = d0e0wP 2d0 =
d0e0zPj = d0grP 2j = d0gi2j = d0gPd0Pv = d0gPe0Pu =d0guP 2e0 = d0gvP 2d0 = d0rkP 2e0 = d0rlP
2d0 = d0imPj =d0jlP j = d0jPe0z = d0k
2Pj = d0kPd0z = d0Pd0Pe0w =d0Pe2
0v = e30P
2u = e20rP
2j = e20i
2j = e20Pd0Pv = e2
0Pe0Pu =e20uP 2e0 = e2
0vP 2d0 = e0grP 2i = e0gi3 = e0gPd0Pu =e0guP 2d0 = e0rjP
2e0 = e0rkP 2d0 = e0rPe0Pj = e0ilP j =e0iPe0z = e0jkPj = e0jPd0z = e0Pd2
0w = e0Pd0Pe0v =e0Pe2
0u = g2P 3v = griP 2e0 = grjP 2d0 = grPd0Pj = gikPj =giPd0z = gj2Pj = gPd2
0v = gPd0Pe0u = gwP 3e0 = rjPe20 =
rkPd0Pe0 = rlPd20 = rmP 3e0 = i2mPe0 = ijlPe0 = ijmPd0 =
ik2Pe0 = iklPd0 = j2kPe0 = j2lPd0 = jk2Pd0 = kmP 2j =l2P 2j = lmP 2i = lzP 2e0 = mzP 2d0
continued
162 ROBERT R. BRUNER
Stem 106 continued(110) d0x
′P 2i = ix′P 2d0 = jP 3Q1 = B4P4d0 = Q1 P 3j = PjP 2Q1 =
PQ1 P 2j
(001) h0d0P2x16,32 = h0P
2d0x16,32 = h2P3x16,35 = Ph2P
2x16,35 =P 2h2Px16,35 = P 3h2x16,35
(101) gx25,24
(111) d0Pd0R1 = e0P3R2 = Pe0P
2R2 = P 2e0PR2 = R2 P 3e0 =P 2D1P
3d0
30 (10) h0d0ix18,20 = h0d0x′P 2i = h0ix
′P 2d0 = h0jP3Q1 = h0B4P
4d0 =h0Q1 P 3j = h0PjP 2Q1 = h0PQ1 P 2j = h2iP
3Q1 =h2x
′P 3j = h2Pjx18,20 = h2PQ1 P 2i = f0P4Q1 = Ph2iP
2Q1 =Ph2jx18,20 = Ph2x
′P 2j = Ph2Q1 P 2i = iP 2h2PQ1 =iQ1 P 3h2 = jx′P 3h2 = B2P
4j = P 2h2x′Pj = x13,35P
4h2
(01) h20d0P
2x16,32 = h20P
2d0x16,32 = h0h2P3x16,35 = h0Ph2P
2x16,35 =h0P
2h2Px16,35 = h0P3h2x16,35 = h2Ph2P
2x16,32 =h2P
3h2x16,32 = Ph2P2h2x16,32 = x13,34P
4h2
(11) h0d0Pd0R1 = h0e0P3R2 = h0gx25,24 = h0Pe0P
2R2 =h0P
2e0PR2 = h0R2 P 3e0 = h0P2D1P
3d0 = h2d0P3R2 =
h2e0x25,24 = h2Pd0P2R2 = h2P
2d0PR2 = h2P2e0R1 =
h2R2 P 3d0 = d0Ph2P2R2 = d0P
2h2PR2 = d0R1P3d0 =
d0P3h2R2 = e0P
2h2R1 = Ph2Pd0PR2 = Ph2Pe0R1 =Ph2P
2d0R2 = Pd0P2h2R2 = Pd0R1P
2d0
31 (1) h20d0ix18,20 = h2
0d0Pd0R1 = h20d0x
′P 2i = h20e0P
3R2 =h2
0gx25,24 = h20ix
′P 2d0 = h20jP
3Q1 = h20Pe0P
2R2 =h2
0B4P4d0 = h2
0Q1 P 3j = h20PjP 2Q1 = h2
0P2e0PR2 =
h20R2 P 3e0 = h2
0P2D1P
3d0 = h20PQ1 P 2j = h0h2d0P
3R2 =h0h2e0x25,24 = h0h2iP
3Q1 = h0h2Pd0P2R2 = h0h2x
′P 3j =h0h2Pjx18,20 = h0h2P
2d0PR2 = h0h2P2e0R1 = h0h2R2 P 3d0 =
h0h2PQ1 P 2i = h0d0Ph2P2R2 = h0d0P
2h2PR2 =h0d0R1P
3d0 = h0d0P3h2R2 = h0e0P
2h2R1 = h0f0P4Q1 =
h0Ph2iP2Q1 = h0Ph2jx18,20 = h0Ph2Pd0PR2 =
h0Ph2Pe0R1 = h0Ph2x′P 2j = h0Ph2Q1 P 2i = h0Ph2P
2d0R2 =h0iP
2h2PQ1 = h0iQ1 P 3h2 = h0jx′P 3h2 = h0B2P
4j =h0Pd0P
2h2R2 = h0Pd0R1P2d0 = h0P
2h2x′Pj = h0x13,35P
4h2 =h1d0Pd0x18,20 = h1d0x
′P 3d0 = h1e0P4Q1 = h1gP 4x′ =
h1Pd0x′P 2d0 = h1Pe0P
3Q1 = h1Q1 P 4e0 = h1B21P4d0 =
h1P2e0P
2Q1 = h1PQ1 P 3e0 = h22d0x25,24 = h2
2P2d0R1 =
h2d0P2h2R1 = h2f0P
4x′ = h2Ph2ix18,20 = h2Ph2Pd0R1 =h2Ph2x
′P 2i = h2ix′P 3h2 = h2B2P
4i = h2B4P5h2 = h2R1P
4e0 =h2P
2D1P4h2 = d2
0Ph1x18,20 = d20x
′P 3h1 = d0Ph1x′P 2d0 =
(continued)
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 163
Stem 106 continued31 (1) (continued) = d0Ph2
2R1 = d0B1P4d0 = d0Pd0P
2h1x′ =
d0B21P4h1 = e0Ph1P
3Q1 = e0P2h1P
2Q1 = e0R1P4h2 =
e0Q1 P 4h1 = e0P3h1PQ1 = f0P
2h2x18,20 = f0x′P 4h2 =
gP 2h1x18,20 = gx′P 4h1 = Ph1Pd20x
′ = Ph1Pe0P2Q1 =
Ph1Q1 P 3e0 = Ph1B21P3d0 = Ph1P
2e0PQ1 = Ph2iP2h2x
′ =Ph2B4P
4h2 = Ph2R1P3e0 = Ph2P
3h2P2D1 = iB2P
4h2 =B1Pd0P
3d0 = B1P2d2
0 = B2P2h2P
2i = Pd0B21P3h1 =
Pe0P2h1PQ1 = Pe0R1P
3h2 = Pe0Q1 P 3h1 = P 2h1Q1 P 2e0 =P 2h1B21P
2d0 = P 2h22P
2D1 = P 2h2B4P3h2 = P 2h2R1P
2e0 =B23P
5h1
32 (1) d60Pd0 = d4
0e0P2e0 = d4
0gP 2d0 = d40Pe2
0 = d30e
20P
2d0 =d30e0Pd0Pe0 = d3
0gPd20 = d2
0e20Pd2
0 = d20e0gP 3e0 = d2
0g2P 3d0 =
d20gPe0P
2e0 = d20iP
2u = d20uP 2i = d0e
30P
3e0 = d0e20gP 3d0 =
d0e20Pe0P
2e0 = d0e0gPd0P2e0 = d0e0gPe0P
2d0 = d0e0Pe30 =
d0g2Pd0P
2d0 = d0gPd0Pe20 = d0riP
2i = d0i4 = d0iPd0Pu =
d0iuP 2d0 = d0jP3v = d0kP 3u = d0vP 3j = d0PjP 2v =
d0PvP 2j = e40P
3d0 = e30Pd0P
2e0 = e30Pe0P
2d0 = e20gPd0P
2d0 =e20Pd0Pe2
0 = e0g2P 4e0 = e0gPd2
0Pe0 = e0iP3v = e0jP
3u =e0uP 3j = e0PjP 2u = e0PuP 2j = e0PvP 2i = g3P 4d0 =g2Pd3
0 = g2Pe0P3e0 = g2P 2e2
0 = giP 3u = gPuP 2i = r2P 4d0 =ri2P 2d0 = rjP 3j = rPjP 2j = i2jP j = iPd2
0u = iPe0P2v =
ivP 3e0 = iwP 3d0 = izP 2j = iP 2e0Pv = jPd0P2v = jPe0P
2u =juP 3e0 = jvP 3d0 = jzP 2i = jP 2d0Pv = jP 2e0Pu = kPd0P
2u =kuP 3d0 = kP 2d0Pu = lP 4v = mP 4u = Pd0vP 2j = Pd0wP 2i =Pd0PjPv = Pe0uP 2j = Pe0vP 2i = Pe0PjPu = uPjP 2e0 =vPjP 2d0
164 ROBERT R. BRUNER
Stem 1075 (1) h4g3 = h6g2
6 (1) h0h4g3 = h0h6g2
7 (10) x7,93
(01) h20h4g3 = h2
0h6g2 = h3h6x
8 (1) h0x7,93
9 (1) x9,102
10 (1) h0x9,102 = h2x9,97
11 (010) h21x9,99
(001) h20x9,102 = h0h2x9,97 = h3x10,82
(101) h1h3x9,86 = d1x7,53 = g2x7,33
(011) h6d0r
(111) D3B1
12 (1) x12,85
13 (1) x13,85
14 (1) gx10,60 = nx9,51 = H1w = rx8,57 = A′N = AN
16 (10) d0x12,60
(01) h1Ph1x10,76 = Gx′ = B1X1
(11) e0x12,55
17 (10) g2x9,39 = grQ2 = x8,33w
(01) h0d0x12,60 = h0e0x12,55 = h2d0x12,55 = h5x16,32 = h6d0P2d0 =
h6Pd20 = Ph2x12,64 = B2B21
18 (10) x18,72
(01) h20d0x12,60 = h2
0e0x12,55 = h0h2d0x12,55 = h0h5x16,32 =h0h6d0P
2d0 = h0h6Pd20 = h0Ph2x12,64 = h0B2B21 = h2
2Px12,60 =h2h6P
3e0 = h2Ph2x12,60 = h2B2Q1 = h6e0P3h2 = h6Ph2P
2e0 =h6Pe0P
2h2 = Pc0x11,61
19 (010) e0rB4 = gx15,47 = vB23 = wx10,27 = wx10,28
(001) h0x18,72 = h2x18,68
(101) P 2x11,61
20 (10) x′R1
(01) h20x18,72 = h0h2x18,68 = h2
2x18,63 = Ph2x15,58
(11) h0P2x11,61
21 (010) h1x′2 = P 2h1x12,55
(110) d0x17,57 = e0x17,52 = gx17,50
(001) h20P
2x11,61 = h0x′R1
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 165
Stem 107 continued22 (100) e0g
3r = e0gm2 = e0r3 = e0w
2 = g2lm = gvw = rlw = rmv
(010) d30B23 = d2
0e0x10,27 = d20e0x10,28 = d2
0gB21 = d0e20B21 =
d0e0gQ1 = d0g2x′ = e3
0Q1 = e20gx′ = e0Pe0B23 = gPd0B23 =
gPe0x10,27 = gPe0x10,28 = ix15,43 = uP 2D1 = vR2 = B4Pu
(001) h30P
2x11,61 = h20x
′R1 = xR1 = ix15,42 = B4Q
(011) rx16,33
23 (1) h40P
2x11,61 = h30x
′R1 = h0xR1 = h0rx16,33 = h0ix15,42 =h0B4Q = riR1 = i2X1 = G21 P 2i
24 (10) d20x16,35 = d0e0x16,32 = gPx16,35 = Pd0x16,42 = Pe0x16,38
(01) h50P
2x11,61 = h40x
′R1 = h20xR1 = h2
0rx16,33 = h20ix15,42 =
h20B4Q = h0riR1 = h0i
2X1 = h0G21 P 2i = h3rR1 = h3ix16,33 =h3x
′Q = yx18,20
25 (100) d30gw = d3
0rm = d20e
20w = d2
0e0gv = d20e0rl = d2
0g2u = d2
0grk =d0e
30v = d0e
20gu = d0e
20rk = d0e0grj = d0e0mz = d0g
2ri =d0glz = d0im
2 = d0jlm = d0k2m = d0kl2 = e4
0u = e30rj =
e20gri = e2
0lz = e0g2Pv = e0gkz = e0gPe0w = e0rmPe0 =
e0ilm = e0jkm = e0jl2 = e0k
2l = g3Pu = g2jz = g2Pd0w =g2Pe0v = grlPe0 = grmPd0 = gikm = gil2 = gj2m = gjkl =gk3 = r2Pu = iuw = iv2 = juv = ku2
(001) h60P
2x11,61 = h50x
′R1 = h30xR1 = h3
0rx16,33 = h30ix15,42 =
h30B4Q = h2
0riR1 = h20i
2X1 = h20G21 P 2i = h0h3rR1 =
h0h3ix16,33 = h0h3x′Q = h0yx18,20 = h3i
2R1 = h3X1P2i =
h4P2x16,32 = xiQ = r2Q
(011) d30R2 = d2
0iQ1 = d20jx
′ = d0e0ix′ = d0gPR2 = d0kPQ1 =
d0Pd0x13,35 = d0Pe0P2D1 = d0B4P
2e0 = d0B21Pj = e20PR2 =
e0gR1 = e0jPQ1 = e0Pd0P2D1 = e0Pe0R2 = e0B4P
2d0 =e0Q1 Pj = giPQ1 = gPd0R2 = gx′Pj = iPd0x10,27 =iPd0x10,28 = iPe0B21 = jPd0B21 = jPe0Q1 = kPd0Q1 =kPe0x
′ = lPd0x′ = mP 2Q1 = Pd0Pe0B4 = B23P
2j
continued
166 ROBERT R. BRUNER
Stem 107 continued26 (1) h7
0P2x11,61 = h6
0x′R1 = h4
0xR1 = h40rx16,33 = h4
0ix15,42 =h4
0B4Q = h30riR1 = h3
0i2X1 = h3
0G21 P 2i = h20h3rR1 =
h20h3ix16,33 = h2
0h3x′Q = h2
0yx18,20 = h0h3i2R1 = h0h3X1P
2i =h0h4P
2x16,32 = h0d30R2 = h0d
20iQ1 = h0d
20jx
′ = h0d0e0ix′ =
h0d0gPR2 = h0d0kPQ1 = h0d0Pd0x13,35 = h0d0Pe0P2D1 =
h0d0B4P2e0 = h0d0B21Pj = h0e
20PR2 = h0e0gR1 =
h0e0jPQ1 = h0e0Pd0P2D1 = h0e0Pe0R2 = h0e0B4P
2d0 =h0e0Q1 Pj = h0giPQ1 = h0gPd0R2 = h0gx′Pj =h0xiQ = h0r
2Q = h0iPd0x10,27 = h0iPd0x10,28 = h0iPe0B21 =h0jPd0B21 = h0jPe0Q1 = h0kPd0Q1 = h0kPe0x
′ =h0lPd0x
′ = h0mP 2Q1 = h0Pd0Pe0B4 = h0B23P2j =
h2d20ix
′ = h2d0e0PR2 = h2d0gR1 = h2d0jPQ1 =h2d0Pd0P
2D1 = h2d0Pe0R2 = h2d0B4P2d0 = h2d0Q1 Pj =
h2e20R1 = h2e0iPQ1 = h2e0Pd0R2 = h2e0x
′Pj =h2iPd0B21 = h2iPe0Q1 = h2jPd0Q1 = h2jPe0x
′ =h2kPd0x
′ = h2lP2Q1 = h2mx18,20 = h2Pd2
0B4 = h2x10,27P2j =
h2x10,28P2j = h2B23P
2i = h2P2e0x13,35 = h2
3iR1 = h4Pd0R1 =d20Ph2P
2D1 = d20Pe0R1 = d2
0P2h2B4 = d0e0Ph2R2 =
d0e0Pd0R1 = d0f0Pd0x′ = d0Ph2iB21 = d0Ph2jQ1 =
d0Ph2kx′ = d0Ph2Pd0B4 = d0iB2Pd0 = e0f0P2Q1 =
e0Ph2iQ1 = e0Ph2jx′ = e0B2P
2j = e0P2h2x13,35 = f0gx18,20 =
f0Pe0PQ1 = f0Q1 P 2e0 = f0B21P2d0 = gPh2ix
′ =gB2P
2i = gP 2h2P2D1 = gB4P
3h2 = gR1P2e0 = Ph2lPQ1 =
Ph2Pe0x13,35 = Ph2x10,27Pj = Ph2x10,28Pj = D2P4e0 =
AP 4d0 = iP 2h2B23 = jB2P2e0 = jP 2h2x10,27 = jP 2h2x10,28 =
kB2P2d0 = kP 2h2B21 = lP 2h2Q1 = mP 2h2x
′ = B2Pe0Pj =Q2P
3j = B5 P 3e0 = PD2P3e0 = PAP 3d0
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 167
Stem 107 continued27 (1) h8
0P2x11,61 = h7
0x′R1 = h5
0xR1 = h50rx16,33 = h5
0ix15,42 =h5
0B4Q = h40riR1 = h4
0i2X1 = h4
0G21 P 2i = h30h3rR1 =
h30h3ix16,33 = h3
0h3x′Q = h3
0yx18,20 = h20h3i
2R1 =h2
0h3X1P2i = h2
0h4P2x16,32 = h2
0d30R2 = h2
0d20iQ1 =
h20d
20jx
′ = h20d0e0ix
′ = h20d0gPR2 = h2
0d0kPQ1 =h2
0d0Pd0x13,35 = h20d0Pe0P
2D1 = h20d0B4P
2e0 = h20d0B21Pj =
h20e
20PR2 = h2
0e0gR1 = h20e0jPQ1 = h2
0e0Pd0P2D1 =
h20e0Pe0R2 = h2
0e0B4P2d0 = h2
0e0Q1 Pj = h20giPQ1 =
h20gPd0R2 = h2
0gx′Pj = h20xiQ = h2
0r2Q = h2
0iPd0x10,27 =h2
0iPd0x10,28 = h20iPe0B21 = h2
0jPd0B21 = h20jPe0Q1 =
h20kPd0Q1 = h2
0kPe0x′ = h2
0lPd0x′ = h2
0mP 2Q1 =h2
0Pd0Pe0B4 = h20B23P
2j = h0h2d20ix
′ = h0h2d0e0PR2 =h0h2d0gR1 = h0h2d0jPQ1 = h0h2d0Pd0P
2D1 =h0h2d0Pe0R2 = h0h2d0B4P
2d0 = h0h2d0Q1 Pj = h0h2e20R1 =
h0h2e0iPQ1 = h0h2e0Pd0R2 = h0h2e0x′Pj = h0h2iPd0B21 =
h0h2iPe0Q1 = h0h2jPd0Q1 = h0h2jPe0x′ = h0h2kPd0x
′ =h0h2lP
2Q1 = h0h2mx18,20 = h0h2Pd20B4 = h0h2x10,27P
2j =h0h2x10,28P
2j = h0h2B23P2i = h0h2P
2e0x13,35 = h0h23iR1 =
h0h4Pd0R1 = h0d20Ph2P
2D1 = h0d20Pe0R1 = h0d
20P
2h2B4 =h0d0e0Ph2R2 = h0d0e0Pd0R1 = h0d0f0Pd0x
′ = h0d0Ph2iB21 =h0d0Ph2jQ1 = h0d0Ph2kx′ = h0d0Ph2Pd0B4 = h0d0iB2Pd0 =h0e0f0P
2Q1 = h0e0Ph2iQ1 = h0e0Ph2jx′ = h0e0B2P
2j =h0e0P
2h2x13,35 = h0f0gx18,20 = h0f0Pe0PQ1 = h0f0Q1 P 2e0 =h0f0B21P
2d0 = h0gPh2ix′ = h0gB2P
2i = h0gP 2h2P2D1 =
h0gB4P3h2 = h0gR1P
2e0 = h0Ph2lPQ1 = h0Ph2Pe0x13,35 =h0Ph2x10,27Pj = h0Ph2x10,28Pj = h0D2P
4e0 = h0AP 4d0 =h0iP
2h2B23 = h0jB2P2e0 = h0jP
2h2x10,27 = h0jP2h2x10,28 =
h0kB2P2d0 = h0kP 2h2B21 = h0lP
2h2Q1 = h0mP 2h2x′ =
h0B2Pe0Pj = h0Q2P3j = h0B5 P 3e0 = h0PD2P
3e0 =h0PAP 3d0 = h1d
30PQ1 = h1d
20Pd0Q1 = h1d
20Pe0x
′ =h1d0e0Pd0x
′ = h1d0gP 2Q1 = h1d0B21P2e0 = h1d0x10,27P
2d0 =h1d0x10,28P
2d0 = h1e20P
2Q1 = h1e0gx18,20 = h1e0Pe0PQ1 =h1e0Q1 P 2e0 = h1e0B21P
2d0 = h1gPd0PQ1 = h1gx′P 2e0 =h1gQ1 P 2d0 = h1Pd2
0x10,27 = h1Pd20x10,28 = h1Pd0Pe0B21 =
h1Pe20Q1 = h1B23P
3e0 = h22d
20PR2 = h2
2d0e0R1 = h22d0iPQ1 =
h22d0Pd0R2 = h2
2d0x′Pj = h2
2gP 2R2 = h22iPd0Q1 = h2
2iPe0x′ =
h22jPd0x
′ = h22kP 2Q1 = h2
2lx18,20 = h22B4P
3e0 = h22B21P
2j =h2
2x10,27P2i = h2
2x10,28P2i = h2
2P2d0x13,35 = h2
2P2e0P
2D1 =h2h4P
3R2 = h2d20Ph2R2 = (continued)
168 ROBERT R. BRUNER
Stem 107 continued27 (1) (continued) = h2d
20Pd0R1 = h2d0f0P
2Q1 = h2d0Ph2iQ1 =h2d0Ph2jx
′ = h2d0B2P2j = h2d0P
2h2x13,35 = h2e0f0x18,20 =h2e0Ph2ix
′ = h2e0B2P2i = h2e0P
2h2P2D1 = h2e0B4P
3h2 =h2e0R1P
2e0 = h2f0Pd0PQ1 = h2f0x′P 2e0 = h2f0Q1 P 2d0 =
h2gPh2PR2 = h2gP 2h2R2 = h2gR1P2d0 = h2Ph2kPQ1 =
h2Ph2Pd0x13,35 = h2Ph2Pe0P2D1 = h2Ph2B4P
2e0 =h2Ph2B21Pj = h2D2P
4d0 = h2iB2P2e0 = h2iP
2h2x10,27 =h2iP
2h2x10,28 = h2jB2P2d0 = h2jP
2h2B21 = h2kP 2h2Q1 =h2lP
2h2x′ = h2B2Pd0Pj = h2Pe2
0R1 = h2Pe0P2h2B4 =
h2B5 P 3d0 = h2PD2P3d0 = h2
3R1P2i = h3riQ =
h4Ph2P2R2 = h4P
2h2PR2 = h4R1P3d0 = h4P
3h2R2 =c0rx18,20 = c0i
2x′ = c0jPR2 = c0kR1 = c0Pd0x16,33 =c0B4P
2i = c0PjR2 = d30Ph1Q1 = d3
0Ph2R1 =d20e0Ph1x
′ = d20B1P
2e0 = d20P
2h1x10,27 = d20P
2h1x10,28 =d0e0B1P
2d0 = d0e0P2h1B21 = d0f0Ph2PQ1 = d0f0P
2h2Q1 =d0gPh1PQ1 = d0gP 2h1Q1 = d0gP 2h2R1 = d0Ph1Pd0x10,27 =d0Ph1Pd0x10,28 = d0Ph1Pe0B21 = d0Ph2
2x13,35 = d0Ph2B2Pj =d0D2P
4h2 = d0Pc0x16,33 = d0jB2P2h2 = d0B1Pd0Pe0 =
d0B5 P 3h2 = d0PD2P3h2 = e2
0Ph1PQ1 = e20P
2h1Q1 =e20P
2h2R1 = e0f0P2h2x
′ = e0gP 2h1x′ = e0Ph1Pd0B21 =
e0Ph1Pe0Q1 = e0Ph22P
2D1 = e0Ph2Pe0R1 = e0Ph2P2h2B4 =
e0iB2P2h2 = e0B1Pd2
0 = e0B23P3h1 = f0Ph2Pd0Q1 =
f0Ph2Pe0x′ = f0B2P
3e0 = f0x10,27P3h2 = f0x10,28P
3h2 =gPh1Pd0Q1 = gPh1Pe0x
′ = gPh22R2 = gPh2Pd0R1 =
gB1P3e0 = gx10,27P
3h1 = gx10,28P3h1 = Ph1B23P
2e0 =Ph2
2ix10,27 = Ph22ix10,28 = Ph2
2jB21 = Ph22kQ1 = Ph2
2lx′ =
Ph22Pe0B4 = Ph2D2P
3d0 = Ph2iB2Pe0 = Ph2jB2Pd0 =Ph2Q2P
2i = Ph2B5 P 2d0 = Ph2PD2P2d0 = ryP 2i = rx′P 2c0 =
yi3 = D2Pd0P3h2 = D2P
2h2P2d0 = A′′P 5h2 = Pc0iP
2D1 =Pc0jR2 = iQ2P
3h2 = iB4P2c0 = lB2P
3h2 = Pd0P2h2B5 =
Pd0P2h2PD2 = Pe0P
2h1B23
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 169
Stem 107 continued28 (10) d0iR1 = jP 2R2 = PjPR2 = P 2d0x16,33 = R2 P 2j = P 2D1P
2i
(01) d50e0g = d4
0e30 = d3
0g2Pe0 = d3
0iv = d30ju = d2
0e20gPe0 =
d20e0g
2Pd0 = d20e0iu = d2
0rij = d20kPv = d2
0lPu = d20wPj =
d0e40Pe0 = d0e
30gPd0 = d0e0ri
2 = d0e0jPv = d0e0kPu =d0e0vPj = d0g
3P 2e0 = d0giPv = d0gjPu = d0guPj =d0r
2P 2e0 = d0rkPj = d0rPd0z = d0ikz = d0iPe0w = d0j2z =
d0jPd0w = d0jPe0v = d0kPd0v = d0kPe0u = d0lPd0u =d0mP 2v = e5
0Pd0 = e20g
2P 2e0 = e20iPv = e2
0jPu = e20uPj =
e0g3P 2d0 = e0g
2Pe20 = e0giPu = e0r
2P 2d0 = e0rjPj = e0ijz =e0iPd0w = e0iPe0v = e0jPd0v = e0jPe0u = e0kPd0u =e0lP
2v = e0mP 2u = g3Pd0Pe0 = griP j = gi2z = giPd0v =giPe0u = gjPd0u = gkP 2v = glP 2u = gwP 2j = r2Pd0Pe0 =rikPe0 = rilPd0 = rj2Pe0 = rjkPd0 = rmP 2j = i2jm = i2kl =ij2l = ijk2 = j3k = kwP 2e0 = lP e0Pv = lvP 2e0 = lwP 2d0 =lzP j = mPd0Pv = mPe0Pu = muP 2e0 = mvP 2d0 = Pe0z
2
29 (1) h0d0iR1 = h0jP2R2 = h0PjPR2 = h0P
2d0x16,33 = h0R2 P 2j =h0P
2D1P2i = h1d0P
2x16,32 = h1P2d0x16,32 = h2iP
2R2 =h2PjR1 = h2R2 P 2i = d0P
2h1x16,32 = d0R1P2i = f0P
3R2 =Ph1Pd0x16,32 = Ph2iPR2 = Ph2jR1 = iP 2h2R2 = iR1P
2d0 =P 3h1x16,38 = P 3h2x16,37 = W1P
3d0
31 (1) d60i = d4
0e0Pj = d40jPe0 = d4
0kPd0 = d30e0iPe0 = d3
0e0jPd0 =d30giPd0 = d3
0lP2e0 = d3
0mP 2d0 = d20e
20iPd0 = d2
0e0gP 2j =d20e0kP 2e0 = d2
0e0lP2d0 = d2
0g2P 2i = d2
0gjP 2e0 = d20gkP 2d0 =
d20gPe0Pj = d2
0kPe20 = d2
0lPd0Pe0 = d20mPd2
0 = d0e30P
2j =d0e
20gP 2i = d0e
20jP
2e0 = d0e20kP 2d0 = d0e
20Pe0Pj =
d0e0giP 2e0 = d0e0gjP 2d0 = d0e0gPd0Pj = d0e0jPe20 =
d0e0kPd0Pe0 = d0e0lPd20 = d0e0mP 3e0 = d0g
2iP 2d0 =d0giPe2
0 = d0gjPd0Pe0 = d0gkPd20 = d0glP 3e0 = d0gmP 3d0 =
d0rP3u = d0i
2Pu = d0mPe0P2e0 = e4
0P2i = e3
0iP2e0 =
e30jP
2d0 = e30Pd0Pj = e2
0giP 2d0 = e20iPe2
0 = e20jPd0Pe0 =
e20kPd2
0 = e20lP
3e0 = e20mP 3d0 = e0g
2P 3j = e0giPd0Pe0 =e0gjPd2
0 = e0gkP 3e0 = e0glP 3d0 = e0lP e0P2e0 = e0mPd0P
2e0 =e0mPe0P
2d0 = g2iPd20 = g2jP 3e0 = g2kP 3d0 = g2Pe0P
2j =g2PjP 2e0 = gkPe0P
2e0 = glPd0P2e0 = glPe0P
2d0 =gmPd0P
2d0 = rPd0P2u = ruP 3d0 = rP 2d0Pu = i2Pd0u =
ijP 2v = ikP 2u = ivP 2j = iwP 2i = iP jPv = j2P 2u = juP 2j =jvP 2i = jP jPu = kuP 2i = lP e3
0 = mPd0Pe20 = uPj2 = zP 3v
170 ROBERT R. BRUNER
Stem 1086 (10) x6,89
(01) h1h4g3 = h1h6g2 = h3h6e1
7 (1) h0x6,89
9 (1) h3x8,93
10 (10) h1x9,102 = h2x9,99 = h6w
(01) h0h3x8,93 = g2A′′
(11) h23x8,80
11 (10) x11,91
(01) h20h3x8,93 = h0h
23x8,80 = h0g2A
′′ = h2g2A′ = h3f1A
′ = px7,53 =xx6,47
13 (1) h1x12,85 = h4x12,60 = h6d20e0 = h6gPe0
14 (1) x14,82
15 (10) e0x11,61
(01) x15,74
16 (100) e0rA′ = e0rA = g2G21 = gnQ2 = grD2 = lx9,51 = x7,34w =
x7,40v = Nx8,33
(010) h0e0x11,61 = h2d0x11,61 = f0x12,55 = B2B4
(001) h0x15,74 = h2x15,68 = h6iPd0
17 (100) x17,76
(010) h20e0x11,61 = h0h2d0x11,61 = h0f0x12,55 = h0B2B4 = h1d0x12,60 =
h1e0x12,55 = h22x15,65 = h5x16,33 = B1x10,27 = B1x10,28
(001) h20x15,74 = h0h2x15,68 = h0h6iPd0 = h2h6P
2j = h6Ph2Pj =h6jP
2h2
18 (1) h0x17,76
19 (10) h1x18,72 = d0x15,56
(01) h20x17,76
(11) h3x18,63
20 (10) gx16,48
(01) h30x17,76 = h0h3x18,63 = ix13,46 = R2
1
21 (10) d20gB4 = d0e
20B4 = d0kB23 = d0lx10,27 = d0lx10,28 = d0mB21 =
e0gx13,35 = e0jB23 = e0kx10,27 = e0kx10,28 = e0lB21 = e0mQ1 =g2P 2D1 = giB23 = gjx10,27 = gjx10,28 = gkB21 = glQ1 = gmx′
(01) h40x17,76 = h2
0h3x18,63 = h0ix13,46 = h0R21
22 (10) h21x
′2 = h1P2h1x12,55 = Ph2
1x12,55 = Ph1B1x′ = qx16,32 = uW1 =
X1Pu
(01) h50x17,76 = h3
0h3x18,63 = h20ix13,46 = h2
0R21
(11) X1Q
23 (1) d0x19,49 = ix16,42 = jx16,38 = kx16,35 = lx16,32 = Pe0x15,41
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 171
Stem 108 continued24 (10) d2
0g4 = d2
0gr2 = d20mw = d0e
20g
3 = d0e20r
2 = d0e0lw = d0e0mv =d0gkw = d0glv = d0gmu = d0rkm = d0rl
2 = e40g
2 = e20kw =
e20lv = e2
0mu = e0grz = e0gjw = e0gkv = e0glu = e0rjm =e0rkl = g2iw = g2jv = g2ku = grim = grjl = grk2 = ru2 = lmz
(01) d20x16,37 = d0e0x16,33 = d0rPQ1 = d0ix13,35 = d0jP
2D1 =d0kR2 = d0B4Pj = d0zx′ = e0iP
2D1 = e0jR2 = giR2 =rPd0Q1 = rPe0x
′ = i2x10,27 = i2x10,28 = ijB21 = ikQ1 = ilx′ =iPe0B4 = j2Q1 = jkx′ = jPd0B4 = mPR2
26 (1) d0x22,39
27 (10) d40e0m = d4
0gl = d30e
20l = d3
0e0gk = d30g
2j = d20e
30k = d2
0e20gj =
d20e0g
2i = d20gmPe0 = d2
0rPv = d20uz = d0e
40j = d0e
30gi =
d0e20mPe0 = d0e0glPe0 = d0e0gmPd0 = d0e0rPu = d0g
3Pj =d0g
2kPe0 = d0g2lPd0 = d0r
2Pj = d0riz = d0rPd0v = d0rPe0u =d0ijw = d0ikv = d0ilu = d0j
2v = d0jku = e50i = e3
0lP e0 =e30mPd0 = e2
0g2Pj = e2
0gkPe0 = e20glPd0 = e0g
2jPe0 =e0g
2kPd0 = e0rPd0u = e0i2w = e0ijv = e0iku = e0j
2u =e0zPv = g3iPe0 = g3jPd0 = g2mP 2e0 = grP 2v = gi2v = giju =gzPu = r2iPe0 = r2jPd0 = ri2l = rijk = rj3 = rwP 2e0 =imPv = jlPv = jmPu = jz2 = k2Pv = klPu = kwPj = lvP j =muPj = Pd0wz = Pe0vz
(01) x′P 2u
(11) ux18,20
continued
172 ROBERT R. BRUNER
Stem 108 continued30 (10) d2
0P3Q1 = d0Pd0P
2Q1 = d0x′P 3e0 = d0Q1 P 3d0 =
d0P2d0PQ1 = e0x
′P 3d0 = gP 4Q1 = Pd20PQ1 = Pd0x
′P 2e0 =Pd0Q1 P 2d0 = Pe0x
′P 2d0 = B21P4e0 = x10,27P
4d0 = x10,28P4d0
(01) d40rPd0 = d4
0ik = d40j
2 = d30e0ij = d3
0gi2 = d30lP j = d3
0Pe0z =d20e
20i
2 = d20e0rP
2e0 = d20e0kPj = d2
0e0Pd0z = d20grP 2d0 =
d20gjPj = d2
0rPe20 = d2
0ilPe0 = d20imPd0 = d2
0jkPe0 = d20jlPd0 =
d20k
2Pd0 = d0e20rP
2d0 = d0e20jP j = d0e0giP j = d0e0rPd0Pe0 =
d0e0ikPe0 = d0e0ilPd0 = d0e0j2Pe0 = d0e0jkPd0 = d0e0mP 2j =
d0grPd20 = d0gijPe0 = d0gikPd0 = d0gj2Pd0 = d0glP 2j =
d0gmP 2i = d0gzP 2e0 = d0jmP 2e0 = d0klP 2e0 = d0kmP 2d0 =d0l
2P 2d0 = d0mPe0Pj = d0uP 2u = d0Pu2 = e30iP j = e2
0rPd20 =
e20ijPe0 = e2
0ikPd0 = e20j
2Pd0 = e20lP
2j = e20mP 2i = e2
0zP 2e0 =e0grP 3e0 = e0gi2Pe0 = e0gijPd0 = e0gkP 2j = e0glP 2i =e0gzP 2d0 = e0imP 2e0 = e0jlP
2e0 = e0jmP 2d0 = e0k2P 2e0 =
e0klP 2d0 = e0lP e0Pj = e0mPd0Pj = e0Pe20z = g2rP 3d0 =
g2i2Pd0 = g2jP 2j = g2kP 2i = g2Pj2 = grPe0P2e0 = gilP 2e0 =
gimP 2d0 = gjkP 2e0 = gjlP 2d0 = gk2P 2d0 = gkPe0Pj =glPd0Pj = gPd0Pe0z = riP 2u = ruP 2i = i3u = imPe2
0 =jlPe2
0 = jmPd0Pe0 = k2Pe20 = klPd0Pe0 = kmPd2
0 = l2Pd20 =
lmP 3e0 = m2P 3d0 = Pd0uPu = u2P 2d0 = vP 3v = wP 3u =PvP 2v
(11) d0Pe0x18,20 = e0Pd0x18,20
31 (1) h0d20P
3Q1 = h0d0Pd0P2Q1 = h0d0Pe0x18,20 = h0d0x
′P 3e0 =h0d0Q1 P 3d0 = h0d0P
2d0PQ1 = h0e0Pd0x18,20 = h0e0x′P 3d0 =
h0gP 4Q1 = h0Pd20PQ1 = h0Pd0x
′P 2e0 = h0Pd0Q1 P 2d0 =h0Pe0x
′P 2d0 = h0B21P4e0 = h0x10,27P
4d0 = h0x10,28P4d0 =
h2d0Pd0x18,20 = h2d0x′P 3d0 = h2e0P
4Q1 = h2gP 4x′ =h2Pd0x
′P 2d0 = h2Pe0P3Q1 = h2Q1 P 4e0 = h2B21P
4d0 =h2P
2e0P2Q1 = h2PQ1 P 3e0 = d2
0Ph2x18,20 = d20x
′P 3h2 =d0Ph2x
′P 2d0 = d0B2P4d0 = d0Pd0P
2h2x′ = d0B21P
4h2 =e0Ph2P
3Q1 = e0P2h2P
2Q1 = e0Q1 P 4h2 = e0P3h2PQ1 =
gP 2h2x18,20 = gx′P 4h2 = Ph2Pd20x
′ = Ph2Pe0P2Q1 =
Ph2Q1 P 3e0 = Ph2B21P3d0 = Ph2P
2e0PQ1 = Pc0P2x16,32 =
B2Pd0P3d0 = B2P
2d20 = Pd0B21P
3h2 = Pe0P2h2PQ1 =
Pe0Q1 P 3h2 = P 2h2Q1 P 2e0 = P 2h2B21P2d0 = B23P
5h2 =P 3c0x16,32
32 (1) P 4x16,32
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 173
Stem 1095 (1) D11
8 (1) h6B1
9 (10) h6N
(01) h23x7,79
(11) h4x8,80
10 (10) x10,100
(01) h0h23x7,79 = h0h4x8,80 = h1h3x8,93 = h2g2H1 = h3c2A
′ =h3f1H1 = px6,53 = e1x6,47
12 (10) x12,86
(01) h6d0l = h6e0k = h6gj
13 (1) x13,87
14 (01) x14,84
(11) y14,83
15 (100) h1x14,82 = h6i2 = f0x11,61
(010) h0y14,83
(001) h0x14,84
(101) h2x14,79
16 (1) h20x14,84 = h0h2x14,79 = h1x15,74 = h4x15,56
17 (10) x17,79
(01) x17,80
18 (10) h0x17,79
(01) h0x17,80
19 (100) rx13,42
(010) h20x17,79
(001) h20x17,80
20 (0011) h3x19,58
(0100) d0rB23 = d0mB4 = e0rx10,27 = e0rx10,28 = e0lB4 = grB21 =gkB4
(0010) h30x17,79
(0001) h30x17,80
(1001) x′Q1
21 (100) h0x′Q1 = h2x
′2 = P 2h2x12,55
(010) h0h3x19,58 = iP 3h25
(110) c0x18,63
(001) h40x17,80
(011) h40x17,79
continued
174 ROBERT R. BRUNER
Stem 109 continued22 (100) rx16,35 = jx15,41
(010) h20h3x19,58 = h2
0x′Q1 = h0h2x
′2 = h0iP3h2
5 = h0P2h2x12,55
(110) d0x18,57 = e0x18,55 = gx18,50
(001) h50x17,80
(011) h50x17,79
23 (0011) h60x17,79 = h4x22,39 = x10,27Q
(1000) d0g3m = d0grw = d0r
2m = e20g
2m = e20rw = e0g
3l = e0grv =e0r
2l = g4k = g2ru = gr2k = kmw = l2w = lmv = m2u
(0100) d0rR2 = d0uQ1 = d0vx′ = kx16,37 = lx16,33 = Pd0x15,47 =wPQ1 = zP 2D1 = B21Pv = x10,27Pu
(0010) h30h3x19,58 = h3
0x′Q1 = h2
0h2x′2 = h2
0iP3h2
5 = h20P
2h2x12,55 =
h0rx16,35 = h0jx15,41 = h31x
′2 = h21P
2h1x12,55 = h1Ph21x12,55 =
h1Ph1B1x′ = h1qx16,32 = h1uW1 = h1X1Q = h1X1Pu =
h2rx16,32 = h2ix15,41 = h23x21,43 = Ph1uX1 = xP 2Q1 = GP 2u =
x6,53P4h1 = B2
1P 2h1 = Pe0x15,42 = x10,28Q = Px6,53P3h1
(0110) e0ux′ = riQ1 = rjx′ = ijB4 = Pe0x15,43 = x10,28Pu
(0001) h60x17,80
24 (1) h70x17,79 = h7
0x17,80 = h0h4x22,39 = h0x10,27Q
25 (10) d0Px17,50 = Pd0x17,50
(01) h80x17,79 = h8
0x17,80 = h20h4x22,39 = h2
0x10,27Q
26 (010) d40x
′ = d20e0PQ1 = d2
0Pd0B21 = d20Pe0Q1 = d0e0Pd0Q1 =
d0e0Pe0x′ = d0gPd0x
′ = d0x10,27P2e0 = d0x10,28P
2e0 =d0B23P
2d0 = e20Pd0x
′ = e0gP 2Q1 = e0B21P2e0 =
e0x10,27P2d0 = e0x10,28P
2d0 = gPe0PQ1 = gQ1 P 2e0 =gB21P
2d0 = Pd20B23 = Pd0Pe0x10,27 = Pd0Pe0x10,28 =
Pe20B21 = uR1
(001) h90x17,79 = h9
0x17,80 = h30h4x22,39 = h3
0x10,27Q
(101) d30e0gr = d3
0lm = d20e
30r = d2
0e0km = d20e0l
2 = d20g
2z = d20gjm =
d20gkl = d2
0uv = d0e20gz = d0e
20jm = d0e
20kl = d0e0gim =
d0e0gjl = d0e0gk2 = d0e0u2 = d0g
2rPe0 = d0g2il = d0g
2jk =d0riv = d0rju = d0m
2Pe0 = d0wPv = e40z = e3
0im = e30jl =
e30k
2 = e20grPe0 = e2
0gil = e20gjk = e0g
2rPd0 = e0g2ik = e0g
2j2 =e0riu = e0lmPe0 = e0m
2Pd0 = e0vPv = e0wPu = g3ij =g2mPj = gkmPe0 = gl2Pe0 = glmPd0 = guPv = gvPu = r2ij =rkPv = rlPu = rwPj = iwz = jvz = kuz = Pd0vw = Pe0uw =Pe0v
2
(111) g2x18,20
27 (1) h100 x17,79 = h10
0 x17,80 = h40h4x22,39 = h4
0x10,27Q = h4Q2
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 175
Stem 109 continued28 (010) h1ux18,20 = h1x
′P 2u = Ph1x′Q = Ph1x
′Pu = B1P3u =
P 2h1ux′ = P 3h1x15,42 = P 3h1x15,43
(001) h110 x17,79 = h11
0 x17,80 = h50h4x22,39 = h5
0x10,27Q = h0h4Q2
(101) d0P2x16,35 = Pd0Px16,35 = P 2d0x16,35
(111) e0P2x16,32 = P 2e0x16,32
29 (100) d0iP2Q1 = d0x
′P 2j = d0Q1 P 2i = e0x′P 2i = iPd0PQ1 =
ix′P 2e0 = iQ1 P 2d0 = jx′P 2d0 = kP 3Q1 = Pd0x′Pj =
B4P4e0 = B21P
3j = x13,35P3d0
(010) d50u = d4
0ri = d30e0Pv = d3
0gPu = d30jz = d3
0Pd0w = d30Pe0v =
d20e
20Pu = d2
0e0rPj = d20e0iz = d2
0e0Pd0v = d20e0Pe0u =
d20gPd0u = d2
0rjPe0 = d20rkPd0 = d2
0i2m = d2
0ijl = d20ik
2 =d20j
2k = d0e20Pd0u = d0e0gP 2v = d0e0riPe0 = d0e0rjPd0 =
d0e0i2l = d0e0ijk = d0e0j
3 = d0e0wP 2e0 = d0g2P 2u =
d0griPd0 = d0gi2k = d0gij2 = d0gPe0Pv = d0gvP 2e0 =d0gwP 2d0 = d0gzPj = d0rlP
2e0 = d0rmP 2d0 = d0jmPj =d0klP j = d0kPe0z = d0lPd0z = d0Pe2
0w = e30P
2v = e20gP 2u =
e20riPd0 = e2
0i2k = e2
0ij2 = e2
0Pe0Pv = e20vP 2e0 = e2
0wP 2d0 =e20zPj = e0grP 2j = e0gi2j = e0gPd0Pv = e0gPe0Pu =
e0guP 2e0 = e0gvP 2d0 = e0rkP 2e0 = e0rlP2d0 = e0imPj =
e0jlP j = e0jPe0z = e0k2Pj = e0kPd0z = e0Pd0Pe0w =
e0Pe20v = g2rP 2i = g2i3 = g2Pd0Pu = g2uP 2d0 = grjP 2e0 =
grkP 2d0 = grPe0Pj = gilP j = giPe0z = gjkPj = gjPd0z =gPd2
0w = gPd0Pe0v = gPe20u = rkPe2
0 = rlPd0Pe0 = rmPd20 =
ijmPe0 = iklPe0 = ikmPd0 = il2Pd0 = iuPu = j2lP e0 =j2mPd0 = jk2Pe0 = jklPd0 = k3Pd0 = lmP 2j = m2P 2i =mzP 2e0
(110) d0jx18,20 = e0ix18,20
(001) h120 x17,79 = h12
0 x17,80 = h60h4x22,39 = h6
0x10,27Q = h20h4Q
2 =h0d0P
2x16,35 = h0e0P2x16,32 = h0Pd0Px16,35 = h0P
2d0x16,35 =h0P
2e0x16,32 = h2d0P2x16,32 = h2P
2d0x16,32 = d0P2h2x16,32 =
Ph2Pd0x16,32 = P 3h2x16,38 = x13,34P3d0
(101) d20P
2R2 = d0Pd0PR2 = d0Pe0R1 = d0P2d0R2 = e0Pd0R1 =
gP 3R2 = Pd20R2 = P 2D1P
3e0
continued
176 ROBERT R. BRUNER
Stem 109 continued30 (10) h0d0iP
2Q1 = h0d0jx18,20 = h0d0x′P 2j = h0d0Q1 P 2i =
h0e0ix18,20 = h0e0x′P 2i = h0iPd0PQ1 = h0ix
′P 2e0 =h0iQ1 P 2d0 = h0jx
′P 2d0 = h0kP 3Q1 = h0Pd0x′Pj =
h0B4P4e0 = h0B21P
3j = h0x13,35P3d0 = h2d0ix18,20 =
h2d0x′P 2i = h2ix
′P 2d0 = h2jP3Q1 = h2B4P
4d0 = h2Q1 P 3j =h2PjP 2Q1 = h2PQ1 P 2j = d0iP
2h2x′ = d0B4P
4h2 =f0Pd0x18,20 = f0x
′P 3d0 = Ph2iPd0x′ = Ph2jP
2Q1 =Ph2kx18,20 = Ph2B4P
3d0 = Ph2Q1 P 2j = Ph2B21P2i =
Ph2PjPQ1 = iB2P3d0 = iB21P
3h2 = jP 2h2PQ1 = jQ1 P 3h2 =kx′P 3h2 = B2Pd0P
2i = Pd0B4P3h2 = P 2h2B4P
2d0 =P 2h2Q1 Pj
(01) h130 x17,79 = h13
0 x17,80 = h70h4x22,39 = h7
0x10,27Q = h30h4Q
2 =h2
0d0P2x16,35 = h2
0e0P2x16,32 = h2
0Pd0Px16,35 = h20P
2d0x16,35 =h2
0P2e0x16,32 = h0h2d0P
2x16,32 = h0h2P2d0x16,32 =
h0d0P2h2x16,32 = h0Ph2Pd0x16,32 = h0P
3h2x16,38 =h0x13,34P
3d0 = h22P
3x16,35 = h2Ph2P2x16,35 = h2P
2h2Px16,35 =h2P
3h2x16,35 = Ph22Px16,35 = Ph2P
2h2x16,35
(11) h0d20P
2R2 = h0d0Pd0PR2 = h0d0Pe0R1 = h0d0P2d0R2 =
h0e0Pd0R1 = h0gP 3R2 = h0Pd20R2 = h0P
2D1P3e0 =
h2d0Pd0R1 = h2e0P3R2 = h2gx25,24 = h2Pe0P
2R2 =h2P
2e0PR2 = h2R2 P 3e0 = h2P2D1P
3d0 = d20Ph2R1 =
d0R1P3e0 = d0P
3h2P2D1 = e0Ph2P
2R2 = e0P2h2PR2 =
e0R1P3d0 = e0P
3h2R2 = gP 2h2R1 = Ph2Pe0PR2 =Ph2P
2d0P2D1 = Ph2P
2e0R2 = Pd0P2h2P
2D1 = Pd0R1P2e0 =
Pe0P2h2R2 = Pe0R1P
2d0
31 (1) h20d
20P
2R2 = h20d0iP
2Q1 = h20d0jx18,20 = h2
0d0Pd0PR2 =h2
0d0Pe0R1 = h20d0x
′P 2j = h20d0Q1 P 2i = h2
0d0P2d0R2 =
h20e0ix18,20 = h2
0e0Pd0R1 = h20e0x
′P 2i = h20gP 3R2 =
h20iPd0PQ1 = h2
0ix′P 2e0 = h2
0iQ1 P 2d0 = h20jx
′P 2d0 =h2
0kP 3Q1 = h20Pd2
0R2 = h20Pd0x
′Pj = h20B4P
4e0 =h2
0B21P3j = h2
0P2D1P
3e0 = h20x13,35P
3d0 = h0h2d0ix18,20 =h0h2d0Pd0R1 = h0h2d0x
′P 2i = h0h2e0P3R2 = h0h2gx25,24 =
h0h2ix′P 2d0 = h0h2jP
3Q1 = h0h2Pe0P2R2 = h0h2B4P
4d0 =h0h2Q1 P 3j = h0h2PjP 2Q1 = h0h2P
2e0PR2 =h0h2R2 P 3e0 = h0h2P
2D1P3d0 = h0h2PQ1 P 2j = h0d
20Ph2R1 =
h0d0iP2h2x
′ = h0d0B4P4h2 = h0d0R1P
3e0 = h0d0P3h2P
2D1 =h0e0Ph2P
2R2 = h0e0P2h2PR2 = h0e0R1P
3d0 = h0e0P3h2R2 =
h0f0Pd0x18,20 = h0f0x′P 3d0 = h0gP 2h2R1 = h0Ph2iPd0x
′ =h0Ph2jP
2Q1 = h0Ph2kx18,20 = h0Ph2Pe0PR2 =h0Ph2B4P
3d0 = h0Ph2Q1 P 2j = (continued)
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 177
Stem 109 continued31 (1) (continued) = h0Ph2B21P
2i = h0Ph2PjPQ1 =h0Ph2P
2d0P2D1 = h0Ph2P
2e0R2 = h0iB2P3d0 =
h0iB21P3h2 = h0jP
2h2PQ1 = h0jQ1 P 3h2 = h0kx′P 3h2 =h0B2Pd0P
2i = h0Pd0P2h2P
2D1 = h0Pd0B4P3h2 =
h0Pd0R1P2e0 = h0Pe0P
2h2R2 = h0Pe0R1P2d0 =
h0P2h2B4P
2d0 = h0P2h2Q1 Pj = h1d
20P
3Q1 = h1d0Pd0P2Q1 =
h1d0Pe0x18,20 = h1d0x′P 3e0 = h1d0Q1 P 3d0 = h1d0P
2d0PQ1 =h1e0Pd0x18,20 = h1e0x
′P 3d0 = h1gP 4Q1 = h1Pd20PQ1 =
h1Pd0x′P 2e0 = h1Pd0Q1 P 2d0 = h1Pe0x
′P 2d0 = h1B21P4e0 =
h1x10,27P4d0 = h1x10,28P
4d0 = h22d0P
3R2 = h22e0x25,24 =
h22iP
3Q1 = h22Pd0P
2R2 = h22x
′P 3j = h22Pjx18,20 =
h22P
2d0PR2 = h22P
2e0R1 = h22R2 P 3d0 = h2
2PQ1 P 2i =h2d0Ph2P
2R2 = h2d0P2h2PR2 = h2d0R1P
3d0 =h2d0P
3h2R2 = h2e0P2h2R1 = h2f0P
4Q1 = h2Ph2iP2Q1 =
h2Ph2jx18,20 = h2Ph2Pd0PR2 = h2Ph2Pe0R1 =h2Ph2x
′P 2j = h2Ph2Q1 P 2i = h2Ph2P2d0R2 =
h2iP2h2PQ1 = h2iQ1 P 3h2 = h2jx
′P 3h2 = h2B2P4j =
h2Pd0P2h2R2 = h2Pd0R1P
2d0 = h2P2h2x
′Pj = h2x13,35P4h2 =
d20Ph1P
2Q1 = d20P
2h1PQ1 = d20R1P
3h2 = d20Q1 P 3h1 =
d0e0Ph1x18,20 = d0e0x′P 3h1 = d0Ph1Pd0PQ1 = d0Ph1x
′P 2e0 =d0Ph1Q1 P 2d0 = d0Ph2
2PR2 = d0Ph2P2h2R2 =
d0Ph2R1P2d0 = d0B1P
4e0 = d0Pd0P2h1Q1 = d0Pd0P
2h2R1 =d0Pe0P
2h1x′ = d0x10,27P
4h1 = d0x10,28P4h1 = e0Ph1x
′P 2d0 =e0Ph2
2R1 = e0B1P4d0 = e0Pd0P
2h1x′ = e0B21P
4h1 =f0Ph2P
3Q1 = f0P2h2P
2Q1 = f0Q1 P 4h2 = f0P3h2PQ1 =
gPh1P3Q1 = gP 2h1P
2Q1 = gR1P4h2 = gQ1 P 4h1 =
gP 3h1PQ1 = Ph1Pd20Q1 = Ph1Pd0Pe0x
′ = Ph1B21P3e0 =
Ph1x10,27P3d0 = Ph1x10,28P
3d0 = Ph22iPQ1 = Ph2
2Pd0R2 =Ph2
2x′Pj = Ph2iP
2h2Q1 = Ph2jP2h2x
′ = Ph2B2P3j =
Ph2Pd20R1 = Ph2P
3h2x13,35 = D2P6h2 = Pc0iR1 = jB2P
4h2 =B1Pd0P
3e0 = B1Pe0P3d0 = B1P
2d0P2e0 = B2P
2h2P2j =
B2PjP 3h2 = Pd0x10,27P3h1 = Pd0x10,28P
3h1 = Pe0B21P3h1 =
P 2h1B21P2e0 = P 2h1x10,27P
2d0 = P 2h1x10,28P2d0 =
P 2h22x13,35 = B5 P 5h2 = PD2P
5h2 = P 3c0x16,33
178 ROBERT R. BRUNER
Stem 1106 (1) h2h4g3 = h2h6g2 = h3h6f1 = h2
4e2
7 (1) h0h2h4g3 = h0h2h6g2 = h0h3h6f1 = h0h24e2 = h2
3h6p = c2p′
8 (1) h3x7,90 = h4x7,79
9 (1) h1h6B1
10 (100) x10,102
(001) h2x9,102
(011) h23x8,83 = g2r1
11 (10) h1x10,100 = h4x10,73 = c1x8,75 = g2x7,40 = D1D2
(01) h0h2x9,102 = h22x9,97 = h6e0r
(11) h0h23x8,83 = h0g2r1 = pm1
13 (10) x13,88
(01) h1x12,86
14 (100) h2x13,85 = h5x13,42 = c1x11,61 = f0x10,65
(001) h1x13,87 = h6Pu
(011) h6Q
15 (10) x15,78
(01) h0h6Q
16 (1000) x16,77
(0100) x16,78
(0010) d0x12,64 = e0x12,60 = gx12,55
(0001) h20h6Q
17 (10) h0d0x12,64 = h0e0x12,60 = h0gx12,55 = h2d0x12,60 = h2e0x12,55 =h5x16,35 = h6d0P
2e0 = h6e0P2d0 = h6Pd0Pe0 = B2x10,27 =
B2x10,28 = Q2x′
(01) h30h6Q = h0x16,78
18 (1000) x18,77
(0100) x18,78
(0010) h1x17,79
(0001) h40h6Q = h2
0d0x12,64 = h20e0x12,60 = h2
0gx12,55 = h20x16,78 =
h0h2d0x12,60 = h0h2e0x12,55 = h0h5x16,35 = h0h6d0P2e0 =
h0h6e0P2d0 = h0h6Pd0Pe0 = h0B2x10,27 = h0B2x10,28 =
h0Q2x′ = h2
2d0x12,55 = h2h5x16,32 = h2h6d0P2d0 = h2h6Pd2
0 =h2Ph2x12,64 = h2B2B21 = h5d0R2 = h6d
20P
2h2 = h6d0Ph2Pd0 =h6gP 3h2 = c0x15,65 = d0B
22
19 (1000) grB4 = wB23
(0100) Px15,65
(0010) h0x18,77
(0001) h0x18,78 = h2x18,72 = d0x15,58
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 179
Stem 110 continued20 (100) h0Px15,65 = h2P
2x11,61 = P 2h2x11,61
(010) h20x18,77
(001) h20x18,78 = h0h2x18,72 = h0d0x15,58 = h2
2x18,68
(101) R1Q1
21 (100) e0x17,57 = gx17,52
(010) h20Px15,65 = h0h2P
2x11,61 = h0P2h2x11,61 = h0R1Q1 =
h1x′Q1 = h2x
′R1 = Ph1Px12,60 = B1PQ1 = P 2h1x12,60
(001) h30x18,77
22 (100) g4r = g2m2 = gr3 = gw2 = rmw
(010) d20e0B23 = d2
0gx10,27 = d20gx10,28 = d0e
20x10,27 = d0e
20x10,28 =
d0e0gB21 = d0g2Q1 = e3
0B21 = e20gQ1 = e0g
2x′ = gPe0B23 =rx16,37 = ix15,47 = jx15,43 = ux13,35 = vP 2D1 = wR2 = B4Pv
(001) h40x18,77 = h4x21,43
23 (10) Ph1x18,63
(01) h50x18,77 = h0h4x21,43
24 (10) d20x16,38 = d0e0x16,35 = d0gx16,32 = e2
0x16,32 = ix17,50 =Pd0x16,48 = Pe0x16,42
(01) h60x18,77 = h2
0h4x21,43
25 (100) d20e0gw = d2
0e0rm = d20g
2v = d20grl = d0e
30w = d0e
20gv = d0e
20rl =
d0e0g2u = d0e0grk = d0g
2rj = d0gmz = d0jm2 = d0klm = d0l
3 =e40v = e3
0gu = e30rk = e2
0grj = e20mz = e0g
2ri = e0glz = e0im2 =
e0jlm = e0k2m = e0kl2 = g3Pv = g2kz = g2Pe0w = grmPe0 =
gilm = gjkm = gjl2 = gk2l = r2Pv = ruz = ivw = juw = jv2 =kuv = lu2
(010) d30P
2D1 = d20e0R2 = d2
0iB21 = d20jQ1 = d2
0kx′ = d20Pd0B4 =
d0e0iQ1 = d0e0jx′ = d0gix′ = d0lPQ1 = d0Pe0x13,35 =
d0x10,27Pj = d0x10,28Pj = e20ix
′ = e0gPR2 = e0kPQ1 =e0Pd0x13,35 = e0Pe0P
2D1 = e0B4P2e0 = e0B21Pj = g2R1 =
gjPQ1 = gPd0P2D1 = gPe0R2 = gB4P
2d0 = gQ1 Pj =iPd0B23 = iPe0x10,27 = iPe0x10,28 = jPd0x10,27 = jPd0x10,28 =jPe0B21 = kPd0B21 = kPe0Q1 = lPd0Q1 = lP e0x
′ =mPd0x
′ = Pe20B4
(001) h70x18,77 = h3
0h4x21,43
26 (1) h80x18,77 = h4
0h4x21,43
27 (01) h90x18,77 = h5
0h4x21,43
(11) iPx16,35 = Pjx16,32
continued
180 ROBERT R. BRUNER
Stem 110 continued28 (100) zx18,20
(010) rP 3Q1 = i2PQ1 = ix′Pj = B4P3j = x13,35P
2i
(110) d50g
2 = d40e
20g = d3
0e40 = d3
0iw = d30jv = d3
0ku = d20e0g
2Pe0 =d20e0iv = d2
0e0ju = d20g
3Pd0 = d20giu = d2
0r2Pd0 = d2
0rik =d20rj
2 = d20lPv = d2
0mPu = d20z
2 = d0e30gPe0 = d0e
20g
2Pd0 =d0e
20iu = d0e0rij = d0e0kPv = d0e0lPu = d0e0wPj = d0gri2 =
d0gjPv = d0gkPu = d0gvPj = d0rlP j = d0rPe0z = d0ilz =d0jkz = d0jPe0w = d0kPd0w = d0kPe0v = d0lPd0v =d0lP e0u = d0mPd0u = e5
0Pe0 = e40gPd0 = e2
0ri2 = e2
0jPv =e20kPu = e2
0vPj = e0g3P 2e0 = e0giPv = e0gjPu = e0guPj =
e0r2P 2e0 = e0rkPj = e0rPd0z = e0ikz = e0iPe0w = e0j
2z =e0jPd0w = e0jPe0v = e0kPd0v = e0kPe0u = e0lPd0u =e0mP 2v = g4P 2d0 = g3Pe2
0 = g2iPu = gr2P 2d0 = grjPj =gijz = giPd0w = giPe0v = gjPd0v = gjPe0u = gkPd0u =glP 2v = gmP 2u = r2Pe2
0 = rilPe0 = rimPd0 = rjkPe0 =rjlPd0 = rk2Pd0 = i2km = i2l2 = ij2m = ijkl = ik3 = j3l =j2k2 = lwP 2e0 = mPe0Pv = mvP 2e0 = mwP 2d0 = mzPj
(001) h100 x18,77 = h6
0h4x21,43 = h0iPx16,35 = h0Pjx16,32 = f0P2x16,32 =
Ph2ix16,32 = P 3h2x15,41 = x13,34P2i
(011) d0iPR2 = d0jR1 = e0iR1 = iPd0R2 = kP 2R2 = P 2d0x16,37 =P 2e0x16,33 = P 2D1P
2j
29 (10) h0rP3Q1 = h0i
2PQ1 = h0ix′Pj = h0B4P
3j = h0zx18,20 =h0x13,35P
2i = h21ux18,20 = h2
1x′P 2u = h1Ph1x
′Q = h1Ph1x′Pu =
h1B1P3u = h1P
2h1ux′ = h1P3h1x15,42 = h1P
3h1x15,43 =h3P
3x16,35 = f0ix18,20 = f0x′P 2i = D3P
6h1 = Ph21ux′ =
Ph1qx18,20 = Ph1B1P2u = Ph1P
2h1x15,42 = Ph1P2h1x15,43 =
Ph2rx18,20 = Ph2i2x′ = Ph2B4P
2i = rx′P 3h2 = qx′P 3h1 =iB2P
2i = iB4P3h2 = B1P
2h1Q = B1P2h1Pu = B1uP 3h1 =
PD3P5h1 = X1P
4e0 = P 2D3P4h1
(01) h110 x18,77 = h7
0h4x21,43 = h20iPx16,35 = h2
0Pjx16,32 =h0f0P
2x16,32 = h0Ph2ix16,32 = h0P3h2x15,41 = h0x13,34P
2i =h1e0P
2x16,32 = h1P2e0x16,32 = e0P
2h1x16,32 = Ph1Pe0x16,32 =W1P
3e0
(11) h0d0iPR2 = h0d0jR1 = h0e0iR1 = h0iPd0R2 = h0kP 2R2 =h0P
2d0x16,37 = h0P2e0x16,33 = h0P
2D1P2j = h1d0P
2x16,35 =h1Pd0Px16,35 = h1P
2d0x16,35 = h2d0iR1 = h2jP2R2 =
h2PjPR2 = h2P2d0x16,33 = h2R2 P 2j = h2P
2D1P2i =
d0Ph1Px16,35 = d0P2h1x16,35 = d0P
2h2x16,33 = d0R1P2j =
e0R1P2i = f0Pd0R1 = Ph1Pd0x16,35 = Ph2jPR2 = Ph2kR1 =
Ph2Pd0x16,33 = Ph2PjR2 = Pc0x22,39 = iP 2h2P2D1 =
iR1P2e0 = jP 2h2R2 = jR1P
2d0 = Pd0R1Pj = P 3h1x16,42
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 181
Stem 1115 (1) h3h6c2 = h5e2
8 (1) h6B2
9 (1000) x9,107
(0010) x9,109
(0001) h0h6B2
(0101) h23x7,81 = h4x8,83 = g2n1 = g2Q3
10 (10) h0x9,107
(01) h20h6B2 = h0x9,109 = h2
1h6B1
(11) h3x9,97
11 (10) h1x10,102
(01) h20x9,107 = h0h3x9,97
12 (10) h2x11,91 = f0x8,78 = gx8,75 = CA′ = CA
(01) h30x9,107 = h2
0h3x9,97 = h23x10,76
13 (1) h40x9,107 = h3
0h3x9,97 = h0h23x10,76 = h3x12,80 = h4x12,64 =
h6d20g = h6d0e
20 = g2x9,40 = xx8,51
14 (10) d0x10,76
(01) h1x13,88
15 (1000) gx11,61
(0100) x15,81
(0010) x15,82
(0001) h0d0x10,76 = h21x13,87 = h1h6Q = h1h6Pu = h2x14,82 = h5x14,42 =
h6Ph1u = c0x12,78
16 (0011) h2x15,74 = h6d0Pj = h6iPe0 = h6jPd0 = D2x′
(1000) grA′ = grA = mx9,51 = x7,40w
(0100) h1x15,78
(0010) h0gx11,61 = h2e0x11,61 = f0x12,60
(0001) h0x15,81 = h0x15,82 = h4x15,58
17 (010) h20gx11,61 = h0h2e0x11,61 = h0f0x12,60 = h1d0x12,64 =
h1e0x12,60 = h1gx12,55 = h22d0x11,61 = h2f0x12,55 = h2B2B4 =
h5x16,37 = B1B23
(110) h1x16,77
(001) h20x15,81 = h2
0x15,82 = h0h4x15,58
(011) h0h2x15,74 = h0h6d0Pj = h0h6iPe0 = h0h6jPd0 = h0D2x′ =
h22x15,68 = h2h6iPd0 = h6d0Ph2i = h6f0P
2d0 = h6kP 2h2 = Q2R1
18 (1) h2x17,76
19 (10) h21x17,79 = h3x18,68 = Ph1x14,74
(01) h0h2x17,76 = h1x18,78 = e0x15,56
continued
182 ROBERT R. BRUNER
Stem 111 continued21 (10) g4n = g2tm = gnr2 = gNw = nmw = rtw = rmN
(01) d0e0gB4 = d0lB23 = d0mx10,27 = d0mx10,28 = e30B4 = e0kB23 =
e0lx10,27 = e0lx10,28 = e0mB21 = g2x13,35 = gjB23 = gkx10,27 =gkx10,28 = glB21 = gmQ1
23 (1) d20x15,41 = e0x19,49 = ix16,48 = jx16,42 = kx16,38 = lx16,35 =
mx16,32
24 (100) d0e0g4 = d0e0gr2 = d0e0mw = d0glw = d0gmv = d0rlm = e3
0g3 =
e30r
2 = e20lw = e2
0mv = e0gkw = e0glv = e0gmu = e0rkm =e0rl
2 = g2rz = g2jw = g2kv = g2lu = grjm = grkl = ruv = m2z
(010) d20rx
′ = d20iB4 = d0e0x16,37 = d0gx16,33 = d0jx13,35 = d0kP 2D1 =
d0lR2 = d0zQ1 = e20x16,33 = e0rPQ1 = e0ix13,35 = e0jP
2D1 =e0kR2 = e0B4Pj = e0zx′ = giP 2D1 = gjR2 = rPd0B21 =rPe0Q1 = i2B23 = ijx10,27 = ijx10,28 = ikB21 = ilQ1 = imx′ =j2B21 = jkQ1 = jlx′ = jPe0B4 = k2x′ = kPd0B4
(001) h1Ph1x18,63 = P 2h1x15,56
26 (1) d0Px18,50 = e0x22,39 = Pd0x18,50
27 (10) d40gm = d3
0e20m = d3
0e0gl = d30g
2k = d30ru = d2
0e30l = d2
0e20gk =
d20e0g
2j = d20g
3i = d20r
2i = d20vz = d0e
40k = d0e
30gj = d0e
20g
2i =d0e0gmPe0 = d0e0rPv = d0e0uz = d0g
2lP e0 = d0g2mPd0 =
d0grPu = d0rjz = d0rPd0w = d0rPe0v = d0ikw = d0ilv =d0imu = d0j
2w = d0jkv = d0jlu = d0k2u = e5
0j = e40gi =
e30mPe0 = e2
0glPe0 = e20gmPd0 = e2
0rPu = e0g3Pj = e0g
2kPe0 =e0g
2lPd0 = e0r2Pj = e0riz = e0rPd0v = e0rPe0u = e0ijw =
e0ikv = e0ilu = e0j2v = e0jku = g3jPe0 = g3kPd0 = grPd0u =
gi2w = gijv = giku = gj2u = gzPv = r2jPe0 = r2kPd0 =ri2m = rijl = rik2 = rj2k = jmPv = klPv = kmPu = kz2 =l2Pu = lwPj = mvPj = Pe0wz
(01) rP 2R2 = i2R2 = uP 2Q1 = zR1 = x′P 2v = Q1 P 2u =Pjx16,33 = PuPQ1
(11) vx18,20
29 (1) P 3x17,50
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 183
Stem 1127 (1) x7,97
8 (100) x8,105
(001) h0x7,97
(111) h3x7,92 = h4x7,81 = g2d2
10 (10) x10,107
(01) h1x9,107 = h3x9,99
11 (10) h4x10,76
(01) h0x10,107 = h2x10,100 = c1x8,78 = e0x7,79 = H1C
12 (1) h0h4x10,76 = h21x10,102 = h6d0m = h6e0l = h6gk = c0x9,97 =
g2G21
13 (10) x13,91
(01) h2x12,86
14 (10) gx10,65
(01) x14,91
15 (1000) gnA′ = gnA = gH1r = tx9,51 = mx8,57 = x7,40N
(0100) h2y14,83
(0010) h1d0x10,76 = h6ij = c0x12,80
(0001) h0x14,91
(0101) h2x14,84
16 (0011) h0h2y14,83 = gx12,58 = d1x12,44 = D2R1
(0100) Px12,80
(0010) h0h2x14,84 = h1x15,82 = h22x14,79
(0001) h20x14,91 = h1x15,81
(1001) h3x15,68
17 (10) h0h3x15,68
(01) h0Px12,80 = h21x15,78 = Ph1x12,78
18 (100) grx8,33 = gmQ2 = wx9,39
(010) x18,83
(001) h20h3x15,68 = h2
1x16,77 = d1x14,42 = qx12,44
20 (01) Pd0x12,55 = x′B21 = Q12
(11) e0rB23 = e0mB4 = grx10,27 = grx10,28 = glB4
21 (1) h0Pd0x12,55 = h0x′B21 = h0Q1
2 = h2x′Q1 = c0x18,68 =
Ph2Px12,60 = B2PQ1 = P 2h2x12,60
22 (10) Px18,68
(01) d0x18,60 = e0x18,57 = gx18,55 = rx16,38 = kx15,41
continued
184 ROBERT R. BRUNER
Stem 112 continued23 (010) d2
0x15,43 = d0rP2D1 = d0uB21 = d0vQ1 = d0wx′ = e0rR2 =
e0uQ1 = e0vx′ = gux′ = riB21 = rjQ1 = rkx′ = rPd0B4 =ikB4 = j2B4 = lx16,37 = mx16,33 = Pe0x15,47 = zx13,35 =x10,27Pv = x10,28Pv = B23Pu
(110) e0g3m = e0grw = e0r
2m = g4l = g2rv = gr2l = lmw = m2v
(001) h0Px18,68 = Ph2x18,63
24 (1) h20Px18,68 = h0Ph2x18,63
25 (10) e0Px17,50 = ix18,50 = Pe0x17,50
(01) h30Px18,68 = h2
0Ph2x18,63 = h21Ph1x18,63 = h1P
2h1x15,56 =e0x21,43 = Ph2
1x15,56
(11) d0Px17,52 = Pd0x17,52
26 (10) d30g
2r = d30m
2 = d20e
20gr = d2
0e0lm = d20gkm = d2
0gl2 = d20uw =
d20v
2 = d0e40r = d0e
20km = d0e
20l
2 = d0e0g2z = d0e0gjm =
d0e0gkl = d0e0uv = d0g2im = d0g
2jl = d0g2k2 = d0gu2 =
d0riw = d0rjv = d0rku = e30gz = e3
0jm = e30kl = e2
0gim = e20gjl =
e20gk2 = e2
0u2 = e0g
2rPe0 = e0g2il = e0g
2jk = e0riv = e0rju =e0m
2Pe0 = e0wPv = g3rPd0 = g3ik = g3j2 = griu = glmPe0 =gm2Pd0 = gvPv = gwPu = r3Pd0 = r2ik = r2j2 = rlPv =rmPu = rz2 = jwz = kvz = luz = Pd0w
2 = Pe0vw
(01) d40Q1 = d3
0e0x′ = d2
0gPQ1 = d20Pd0x10,27 = d2
0Pd0x10,28 =d20Pe0B21 = d0e
20PQ1 = d0e0Pd0B21 = d0e0Pe0Q1 =
d0gPd0Q1 = d0gPe0x′ = d0B23P
2e0 = e20Pd0Q1 = e2
0Pe0x′ =
e0gPd0x′ = e0x10,27P
2e0 = e0x10,28P2e0 = e0B23P
2d0 =g2P 2Q1 = gB21P
2e0 = gx10,27P2d0 = gx10,28P
2d0 =Pd0Pe0B23 = Pe2
0x10,27 = Pe20x10,28 = uPR2 = vR1 = PuR2
28 (1) d0Pd0x16,32 = e0P2x16,35 = gP 2x16,32 = Pe0Px16,35 =
P 2d0x16,38 = P 2e0x16,35
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 185
Stem 1137 (01) h6C
(11) h3C1
9 (1) x9,111
10 (1) x10,109
11 (10) h6gr
(01) h21x9,107 = h1h3x9,99 = h2
3x9,86 = c1x8,80 = e1x7,53
(11) h1x10,107 = h22x9,102 = f0x7,79 = D1A
′ = D1A
12 (1) gx8,78
13 (10) x13,93
(01) Px9,99
14 (10) h2x13,88
(01) h0Px9,99 = h1x13,91 = h6Pv = Ph1x9,97
15 (01) h1gx10,65 = h22x13,85 = h2h5x13,42 = h2c1x11,61 = h2f0x10,65 =
h4x14,67
(11) h3x14,79
16 (01) h0h3x14,79
(11) e0x12,64 = gx12,60
17 (0011) Q2Q1
(1011) h2x16,77
(0100) e0gx9,51 = e0rx7,40 = e0A′m = e0Am = gnx8,33 = grx7,34 =
gtQ2 = gD2m = gA′l = gAl = Nx9,39 = G21 w
(0010) h0e0x12,64 = h0gx12,60 = h1Px12,80 = h2d0x12,64 = h2e0x12,60 =h2gx12,55 = h5x16,38 = h6d
20Pd0 = h6e0P
2e0 = h6gP 2d0 =h6Pe2
0 = Ph1x12,80 = Pc0x10,76 = B2B23 = B3x′
(1010) h2x16,78
(0001) h20h3x14,79 = h1h3x15,68
18 (10) x18,85
(01) P 2x10,76
19 (1000) Pd0x11,61
(0100) h1x18,83
(0010) h0x18,85 = h2x18,78 = e0x15,58
(0001) h0P2x10,76
(1001) ix12,55 = B4x′
20 (0011) h20x18,85 = h0h2x18,78 = h0e0x15,58 = h2
2x18,72 = h2d0x15,58
(1000) h6P3j
(0100) B2R2 = R1B21
(0111) h0Pd0x11,61 = h2Px15,65 = Ph2x15,65
(0110) h0ix12,55 = h0B4x′
(0001) h20P
2x10,76
continued
186 ROBERT R. BRUNER
Stem 113 continued21 (100) gx17,57
(010) h20Pd0x11,61 = h0h2Px15,65 = h0Ph2x15,65 = h0B2R2 =
h0R1B21 = h1Pd0x12,55 = h1x′B21 = h1Q1
2 = h22P
2x11,61 =h2P
2h2x11,61 = h2R1Q1 = d0Ph1x12,55 = d0B1x′ = Ph2
2x11,61 =P 2h1x12,64
(001) h30P
2x10,76 = h3x′2 = xx16,32
(011) h20ix12,55 = h2
0B4x′
(111) rx15,41
22 (100) x22,71
(010) d20gB23 = d0e
20B23 = d0e0gx10,27 = d0e0gx10,28 = d0g
2B21 =d0uB4 = e3
0x10,27 = e30x10,28 = e2
0gB21 = e0g2Q1 = g3x′ =
jx15,47 = kx15,43 = vx13,35 = wP 2D1
(001) h40P
2x10,76 = h30ix12,55 = h3
0B4x′ = h0h3x
′2 = h0xx16,32 =h0rx15,41 = g2x18,20 = xix′
(011) r2x′ = riB4
23 (10) h0x22,71
(01) h50P
2x10,76 = h40ix12,55 = h4
0B4x′ = h2
0h3x′2 = h2
0xx16,32 =h2
0rx15,41 = h0g2x18,20 = h0xix′ = h0r2x′ = h0riB4 = h3rx16,32 =
h3ix15,41 = i2x9,40 = x8,51P2i
(11) h1Px18,68 = Ph1x18,68
24 (10) d20x16,42 = d0e0x16,38 = d0gx16,35 = e2
0x16,35 = e0gx16,32 =ix17,52 = jx17,50 = Pe0x16,48
(01) h60P
2x10,76 = h50ix12,55 = h5
0B4x′ = h3
0h3x′2 = h3
0xx16,32 =h3
0rx15,41 = h20g2x18,20 = h2
0xix′ = h20r
2x′ = h20riB4 = h2
0x22,71 =h0h3rx16,32 = h0h3ix15,41 = h0i
2x9,40 = h0x8,51P2i = h3xx18,20 =
h3rix′ = h3i
2B4 = Ph2x19,58
25 (10) d20g
2w = d20grm = d0e
20gw = d0e
20rm = d0e0g
2v = d0e0grl =d0g
3u = d0g2rk = d0r
2u = d0km2 = d0l2m = e4
0w = e30gv =
e30rl = e2
0g2u = e2
0grk = e0g2rj = e0gmz = e0jm
2 = e0klm =e0l
3 = g3ri = g2lz = gim2 = gjlm = gk2m = gkl2 = r3i = rvz =iw2 = jvw = kuw = kv2 = luv = mu2
(01) d30x13,35 = d2
0e0P2D1 = d2
0gR2 = d20ix10,27 = d2
0ix10,28 =d20jB21 = d2
0kQ1 = d20lx
′ = d20Pe0B4 = d0e
20R2 = d0e0iB21 =
d0e0jQ1 = d0e0kx′ = d0e0Pd0B4 = d0giQ1 = d0gjx′ =d0mPQ1 = d0B23Pj = e2
0iQ1 = e20jx
′ = e0gix′ =e0lPQ1 = e0Pe0x13,35 = e0x10,27Pj = e0x10,28Pj = g2PR2 =gkPQ1 = gPd0x13,35 = gPe0P
2D1 = gB4P2e0 = gB21Pj =
iPe0B23 = jPd0B23 = jPe0x10,27 = jPe0x10,28 = kPd0x10,27 =kPd0x10,28 = kPe0B21 = lPd0B21 = lP e0Q1 = mPd0Q1 =mPe0x
′
27 (1) d0ix16,32 = jPx16,35 = Pjx16,35 = P 2d0x15,41
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 187
Stem 1146 (1) G1
9 (10) x9,112
(11) h2h6B2
10 (10) h2x9,107 = h3x9,102 = h6gn
(01) h1x9,111 = h2x9,109 = c1x7,79 = D1H1
(11) h4x9,86
12 (10) gx8,80
(01) x12,93
14 (100) Q22
(010) e0x10,76
(001) h1Px9,99 = D3x′ = Ph1x9,99
15 (10) x15,90
(01) h0Q22
16 (10) h2gx11,61 = h5x15,41 = h6d20i = h6e0Pj = h6jPe0 = h6kPd0 =
f0x12,64 = D2Q1 = Ax′
(01) h0x15,90 = h2x15,82
17 (1) h0h2gx11,61 = h0h5x15,41 = h0h6d20i = h0h6e0Pj = h0h6jPe0 =
h0h6kPd0 = h0f0x12,64 = h0D2Q1 = h0Ax′ = h1e0x12,64 =h1gx12,60 = h2
2e0x11,61 = h22x15,74 = h2h6d0Pj = h2h6iPe0 =
h2h6jPd0 = h2f0x12,60 = h2D2x′ = h6d0Ph2j = h6e0Ph2i =
h6f0P2e0 = h6lP
2h2 = c0x14,79 = Ph2x12,78 = B2B5 =B2PD2 = B3R1
18 (100) x18,87
(010) ix11,61
(011) Px14,79
19 (0011) h0ix11,61
(1000) h1P2x10,76 = P 2h1x10,76
(1011) X1x′
(0010) h0Px14,79
(0110) h1x18,85 = h22x17,76 = f0x15,58 = gx15,56
(0101) h0x18,87 = B4R1
20 (010) h20Px14,79 = h2
1x18,83 = Ph1x15,68
(101) gx16,54 = nx15,41 = rx14,46 = mx13,42
(011) h20ix11,61 = h2
0x18,87 = h0B4R1 = h0X1x′ = h3P
2x11,61
21 (10) d0g2B4 = d0mB23 = e2
0gB4 = e0lB23 = e0mx10,27 = e0mx10,28 =gkB23 = glx10,27 = glx10,28 = gmB21 = rx15,43
(01) h30ix11,61 = h3
0x18,87 = h20B4R1 = h2
0X1x′ = h0h3P
2x11,61 =h3x
′R1 = g2R1 = xx16,33 = rx15,42
continued
188 ROBERT R. BRUNER
Stem 114 continued22 (1) h4
0ix11,61 = h40x18,87 = h3
0B4R1 = h30X1x
′ = h20h3P
2x11,61 =h0h3x
′R1 = h0g2R1 = h0xx16,33 = h0rx15,42 = xiR1 = r2R1 =riX1 = yx16,32 = i2G21 = x9,40Q
23 (01) h50ix11,61 = h5
0x18,87 = h40B4R1 = h4
0X1x′ = h3
0h3P2x11,61 =
h20h3x
′R1 = h20g2R1 = h2
0xx16,33 = h20rx15,42 = h0xiR1 =
h0r2R1 = h0riX1 = h0yx16,32 = h0i
2G21 = h0x9,40Q =
h1x22,71 = h3xR1 = h3rx16,33 = h3ix15,42 = h3B4Q = c0x′2 =
yix′ = P 2c0x12,55
(11) d0e0x15,41 = gx19,49 = jx16,48 = kx16,42 = lx16,38 = mx16,35
24 (01) d20rQ1 = d2
0jB4 = d0e0rx′ = d0e0iB4 = d0gx16,37 = d0kx13,35 =
d0lP2D1 = d0mR2 = d0zB21 = e2
0x16,37 = e0gx16,33 =e0jx13,35 = e0kP 2D1 = e0lR2 = e0zQ1 = grPQ1 = gix13,35 =gjP 2D1 = gkR2 = gB4Pj = gzx′ = rPd0x10,27 = rPd0x10,28 =rPe0B21 = ijB23 = ikx10,27 = ikx10,28 = ilB21 = imQ1 =j2x10,27 = j2x10,28 = jkB21 = jlQ1 = jmx′ = k2Q1 = klx′ =kPe0B4 = lPd0B4
(11) d0g5 = d0g
2r2 = d0gmw = d0rm2 = e2
0g4 = e2
0gr2 = e20mw =
e0glw = e0gmv = e0rlm = g2kw = g2lv = g2mu = grkm =grl2 = ruw = rv2
26 (1) d0Px18,55 = e0Px18,50 = gx22,39 = Pd0x18,55 = Pe0x18,50
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 189
Stem 1156 (1) h6D1
8 (1) h2x7,97
9 (1) h2x8,105 = h6gd1
10 (1) h1x9,112 = h23x8,93 = g2x6,47 = D3G
11 (1) x11,101
13 (10) x13,95
(01) D2Q2
14 (10) h0x13,95
(01) h0D2Q2 = h22x12,86 = h4x13,73 = gx10,73 = D3R1
15 (10) h21Px9,99 = h1D3x
′ = h1Ph1x9,99 = h3x14,82 = h6rPd0 = h6ik =h6j
2 = c0x12,85 = d0x11,80 = H1x′ = GX1 = x6,53u = B1PD3
(01) h20x13,95 = h2x14,91
16 (10) Px12,85
(01) h1x15,90 = h22y14,83 = h2
2x14,84 = A′R1 = AR1
17 (10) x17,93
(01) x17,94
18 (1) h0x17,94 = h3x17,76
19 (10) x19,86
(01) h20x17,94 = h0h3x17,76 = rx13,46 = ix12,58 = X1R1
20 (1000) grB23 = gmB4
(0100) x′x10,27
(0111) Pe0x12,55 = x′x10,28
(0010) h21P
2x10,76 = h1P2h1x10,76 = h1X1x
′ = Ph21x10,76 = B1W1
(0001) h30x17,94 = h2
0h3x17,76 = h0rx13,46 = h0ix12,58 = h0X1R1 =h2
3x18,63
(0101) d0Px12,60 = Pd0x12,60 = Q1 B21
21 (1) h40x17,94 = h3
0h3x17,76 = h20rx13,46 = h2
0ix12,58 = h20X1R1 =
h0h23x18,63 = h0d0Px12,60 = h0Pd0x12,60 = h0Pe0x12,55 =
h0x′x10,28 = h0Q1 B21 = h2Pd0x12,55 = h2x
′B21 = h2Q12 =
h3ix13,46 = h3R21 = c0x18,72 = d0Ph2x12,55 = d0B2x
′ = G21 Q =P 2h2x12,64
22 (10) d0x18,63
(01) e0x18,60 = gx18,57 = rx16,42 = lx15,41
23 (100) g4m = g2rw = gr2m = m2w
(010) d20x15,47 = d0e0x15,43 = d0rx13,35 = d0ux10,27 = d0ux10,28 =
d0vB21 = d0wQ1 = d0B4z = e0rP2D1 = e0uB21 = e0vQ1 =
e0wx′ = grR2 = guQ1 = gvx′ = rix10,27 = rix10,28 = rjB21 =rkQ1 = rlx′ = rPe0B4 = ilB4 = jkB4 = mx16,37 = B23Pv
(001) h0d0x18,63 = h2Px18,68 = Ph2x18,68
continued
190 ROBERT R. BRUNER
Stem 115 continued24 (1) h2
0d0x18,63 = h0h2Px18,68 = h0Ph2x18,68 = h2Ph2x18,63 =P 2h2x15,58
25 (10) d20x17,50 = e0Px17,52 = gPx17,50 = ix18,55 = jx18,50 =
Pd0x17,57 = Pe0x17,52
(01) Ph1x′2 = B1x18,20 = x12,55P
3h1
(11) ux16,32
Stem 1168 (1) h2h6C
11 (1) h6x′
12 (10) D22
(01) h0h6x′
13 (100) x13,97
(010) h0D22 = h3x12,86
(001) h20h6x
′
14 (1) h1x13,95
15 (1000) h22x13,88 = h3x14,84 = H1R1
(1100) h3y14,83
(0010) x15,96
(0001) x15,97
16 (1000) x16,95
(0100) gx12,64
(0010) h0h3y14,83
(0001) h0x15,96
17 (1000) g2x9,51 = grx7,40 = gA′m = gAm
(0100) h0x16,95
(0010) h0gx12,64 = h1Px12,85 = h2e0x12,64 = h2gx12,60 = h4Px12,60 =h5x16,42 = h6d
20Pe0 = h6d0e0Pd0 = h6gP 2e0 = c0x14,82 =
Ph1x12,85 = Q2B21 = B3Q1 = x7,33x′ = x7,34x
′
(0001) h20x15,96
18 (1000) Px14,82
(0100) h1x17,93 = h22x16,77 = h2
2x16,78 = f0x14,67 = d1x14,46 = rx12,48 =tx12,44 = x8,32R1 = x8,33R1
(0111) h3x17,79
(0010) h20x16,95
(0001) h30x15,96 = h3x17,80
19 (0011) h0h3x17,79
(1000) ix12,60 = jx12,55 = B4Q1
(0010) h30x16,95
(0110) h4x18,63
(0001) h40x15,96 = h0h3x17,80 = h2P
2x10,76 = P 2h2x10,76
(1001) d0x15,65 = Pe0x11,61
(0101) h2x18,85 = gx15,58
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 191
Stem 116 continued20 (0011) h2
0h3x17,79
(1011) h0ix12,60 = h0jx12,55 = h0B4Q1 = h2ix12,55 = h2B4x′
(1100) Ph2x15,68 = B2P2D1 = R1x10,28
(0010) h40x16,95 = rP 3h2
5
(1111) h6d0P2i = h6iP
2d0 = R1x10,27
(1010) h0d0x15,65 = h0Pe0x11,61 = h2Pd0x11,61 = d0Ph2x11,61
(0110) h0h2x18,85 = h0gx15,58 = h22x18,78 = h2e0x15,58 = c0x17,76
(0001) h50x15,96 = h2
0h3x17,80 = h0h2P2x10,76 = h0P
2h2x10,76
(0101) h0h4x18,63
21 (100) h20d0x15,65 = h2
0Pe0x11,61 = h0h2Pd0x11,61 = h0d0Ph2x11,61 =h0Ph2x15,68 = h0B2P
2D1 = h0R1x10,28 = h1d0Px12,60 =h1Pd0x12,60 = h1x
′x10,27 = h1Q1 B21 = h22Px15,65 =
h2Ph2x15,65 = h2B2R2 = h2R1B21 = d0Ph1x12,60 = d0B1Q1 =d0B2R1
(010) h50x16,95 = h0rP
3h25 = h2
3x19,58
(001) h60x15,96 = h3
0h3x17,80 = h20h2P
2x10,76 = h20P
2h2x10,76 =h3
1P2x10,76 = h2
1P2h1x10,76 = h2
1X1x′ = h1Ph2
1x10,76 = h1B1W1 =h3x
′Q1 = D3P2u = Ph1Gx′ = Ph1B1X1 = xx16,35 = qx15,42 =
qx15,43 = PD3Q = PD3Pu = x8,75P3h1 = uP 2D3
(101) h20ix12,60 = h2
0jx12,55 = h20B4Q1 = h0h2ix12,55 = h0h2B4x
′ =h1Pe0x12,55 = h1x
′x10,28 = e0Ph1x12,55 = e0B1x′
(011) h30h3x17,79 = h2
0h4x18,63
(111) h0h6d0P2i = h0h6iP
2d0 = h0R1x10,27 = h2h6P3j = h6Ph2P
2j =h6jP
3h2 = h6P2h2Pj
22 (100) g4t = g2nw = g2rN = gnrm = gr2t = tmw = m2N
(010) x22,78
(001) d0e0gB23 = d0g2x10,27 = d0g
2x10,28 = d0vB4 = e30B23 =
e20gx10,27 = e2
0gx10,28 = e0g2B21 = e0uB4 = g3Q1 = r2Q1 =
rjB4 = kx15,47 = lx15,43 = wx13,35
23 (1) h0x22,78 = h1d0x18,63 = h2x22,71 = d0x19,58 = Ph1x18,72 =Pd0x15,56
24 (1) d20x16,48 = d0e0x16,42 = d0gx16,38 = e2
0x16,38 = e0gx16,35 =g2x16,32 = ix17,57 = jx17,52 = kx17,50
192 ROBERT R. BRUNER
Stem 1177 (1) h2G1 = h6G
10 (10) x10,113
(01) x10,114
11 (100) x11,103
(010) h0x10,113
(001) h0x10,114 = h6R1
12 (1000) x12,96
(0100) h1h6x′
(0010) h20x10,113
(0001) h20x10,114 = h0h6R1
13 (10) h30x10,113
(01) h30x10,114 = h2
0h6R1 = h0x12,96
14 (0010) h40x10,113
(1010) gx10,76
(0110) h3x13,88
(0001) h40x10,114 = h3
0h6R1 = h20x12,96
15 (100) x15,98
(010) h50x10,113 = h2
1x13,95 = h4x14,74 = c0x12,86
(001) h50x10,114 = h4
0h6R1 = h30x12,96 = h3h6Q = h5P
3h25
16 (10000) g2x8,57 = gnx7,40 = gH1m = gtA′ = gtA = rx10,60
(01000) h2x15,90
(00011) h4x15,65 = h6d20j = h6d0e0i = h6gPj = h6kPe0 = h6lPd0 =
D2B21 = AQ1 = A′′x′
(10011) Q2B4
(00100) Px12,86
(00001) h60x10,114 = h5
0h6R1 = h40x12,96 = h0h3h6Q = h0h5P
3h25 = d0x12,78
(00010) h1x15,96 = h1x15,97 = h3x15,78 = c0x13,87
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 193
Stem 117 continued17 (0011) h3x16,78 = x7,34R1
(1000) Px13,87
(0100) h1x16,95
(0010) h0h2x15,90 = h22x15,82 = c0y14,83 = c0x14,84 = c1x14,67 =
e0x13,73 = e1x13,42 = nx12,48 = Cx11,35
(0110) h3x16,77
(0001) h70x10,114 = h6
0h6R1 = h50x12,96 = h2
0h3h6Q = h20h5P
3h25 =
h0h4x15,65 = h0h6d20j = h0h6d0e0i = h0h6gPj = h0h6kPe0 =
h0h6lPd0 = h0d0x12,78 = h0D2B21 = h0AQ1 = h0A′′x′ =
h0Q2B4 = h1gx12,64 = h22gx11,61 = h2h5x15,41 = h2h6d
20i =
h2h6e0Pj = h2h6jPe0 = h2h6kPd0 = h2f0x12,64 = h2D2Q1 =h2Ax′ = h4Ph2x11,61 = h6d0f0Pd0 = h6d0Ph2k = h6e0Ph2j =h6gPh2i = h6mP 2h2 = B2PA = x7,33R1
18 (10) jx11,61
(01) Px14,84
19 (0011) h2Px14,79 = Ph2x14,79
(1000) g3Q2 = grx9,39 = gmx8,33 = r2Q2
(0010) h0jx11,61 = h1Px14,82 = h2ix11,61 = Ph1x14,82
(0110) h2x18,87
(0001) h0Px14,84
(0101) h3x18,77
21 (10) e0g2B4 = e0mB23 = glB23 = gmx10,27 = gmx10,28 = rx15,47
(01) x21,84
23 (1) d0gx15,41 = e20x15,41 = rx17,50 = kx16,48 = lx16,42 = mx16,38
194 ROBERT R. BRUNER
Stem 1189 (10) x9,115
(01) x9,116
10 (100) h3x9,107
(010) x10,116
(001) h0x9,116
11 (100) x11,106
(010) h0h3x9,107 = h23x9,97
(001) h20x9,116 = h0x10,116
12 (100) x12,100
(010) h1x11,103
(001) h30x9,116 = h2
0x10,116
13 (0011) h0x12,100
(1000) AQ2
(0100) h21h6x
′ = h6Ph1B1 = c0x10,102 = d0x9,97
(1010) A′Q2
(0001) h40x9,116 = h3
0x10,116
14 (0011) h20x12,100
(1000) x14,104
(0010) h0A′Q2
(1111) Px10,102
(0001) h50x9,116 = h4
0x10,116 = h2x13,95
15 (10) h1h3x13,88 = H1Q1
(01) h60x9,116 = h5
0x10,116 = h30x12,100 = h0h2x13,95 = h0x14,104
16 (10) d0x12,80
(01) h1x15,98 = h3x15,81 = d1x12,48 = e1x12,44 = Q2X1
17 (1) h1Px12,86 = Ph1x12,86
18 (010) h1Px13,87 = h5x17,50 = h6P2u = c0x15,78 = Ph1x13,87
(110) e0g2A′ = e0g
2A = e0mx7,40 = g3D2 = gnx9,39 = grG21 =gtx8,33 = glx7,40 = gmx7,34 = nrQ2 = r2D2 = vx9,51
(001) h21x16,95 = h1h3x16,77 = h2x17,93 = gx14,67 = d1P
3h25 = e1x14,42 =
qx12,48 = yx12,44 = x8,32Q1
19 (1) Px15,78
20 (10) h2x19,86
(01) d20x12,55 = e0Px12,60 = Pd0x12,64 = Pe0x12,60 = x′B23 =
Q1 x10,27 = Q1 x10,28 = B221
continued
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 195
Stem 118 continued21 (1) h0d
20x12,55 = h0e0Px12,60 = h0Pd0x12,64 = h0Pe0x12,60 =
h0x′B23 = h0Q1 x10,27 = h0Q1 x10,28 = h0B
221 = h2d0Px12,60 =
h2Pd0x12,60 = h2Pe0x12,55 = h2x′x10,27 = h2x
′x10,28 =h2Q1 B21 = h5Px16,35 = h6d0P
3e0 = h6e0P3d0 = h6Pd0P
2e0 =h6Pe0P
2d0 = d0Ph2x12,60 = d0B2Q1 = e0Ph2x12,55 = e0B2x′
22 (100) gx18,60 = rx16,48 = mx15,41
(010) e0x18,63
(001) h1x21,84 = Ph1x17,79
(011) d0x18,68
Stem 1199 (1) x9,117
10 (10) x10,118
(01) h0x9,117 = h1x9,115 = h4x9,97
11 (1000) x11,109
(0010) h1h3x9,107 = h23x9,99 = h6gt = gx7,83 = g2x7,53
(0110) h6Q1
(0001) h1x10,116
12 (100) D2A
(010) h0x11,109
(001) h0h6Q1 = h2h6x′
(111) H1Q2 = D2A′
13 (10000) gx9,86
(00011) h20x11,109
(01000) h4x12,80
(00100) h1x12,100
(00001) h20h6Q1 = h0h2h6x
′
(01001) d0x9,99
(00010) h0D2A = h21x11,103 = h2D
22
14 (10) x14,108
(01) h30h6Q1 = h3
0x11,109 = h20h2h6x
′ = h0d0x9,99 = h31h6x
′ =h1h6Ph1B1 = h1c0x10,102 = h1d0x9,97 = h3x13,91 = h6d0v
15 (100) x15,103
(010) h1Px10,102 = Ph1x10,102
(001) h0x14,108 = h3x14,91 = D2X1 = Q2x8,32
17 (1) h2x16,95 = h23x15,68 = Q2x10,27
18 (01) h0h2x16,95 = h0h23x15,68 = h0Q2x10,27 = pP 3h2
5 = yx12,45
(11) Pd0x10,76
19 (100) ix12,64 = jx12,60 = kx12,55 = B4B21
(010) h4x18,68
(001) h0Pd0x10,76 = h21Px13,87 = h1h5x17,50 = h1h6P
2u = h1c0x15,78 =h1Ph1x13,87 = h2Px14,82 = h3x18,83 = h6Ph1Q = h6Ph1Pu =h6P
2h1u = Ph2x14,82 = Pc0x12,78
(101) d20x11,61 = e0x15,65
(111) d0x15,68
continued
196 ROBERT R. BRUNER
Stem 119 continued20 (0011) h0d
20x11,61 = h0e0x15,65 = h0ix12,64 = h0jx12,60 = h0kx12,55 =
h0B4B21 = h2d0x15,65 = h2ix12,60 = h2jx12,55 = h2Pe0x11,61 =h2B4Q1 = e0Ph2x11,61 = f0Px12,60 = B2x13,35 = x′B5
(1000) x20,91
(0100) h1Px15,78 = Ph1x15,78
(0010) h0d0x15,68 = h6d0P2j = h6e0P
2i = h6iP2e0 = h6jP
2d0 =h6Pd0Pj = Ph2x15,74 = x′PD2 = R1B23
(0001) h0h4x18,68 = h22x18,85 = h2h4x18,63 = h2gx15,58 = B2x13,34
21 (100) g2x13,42 = nx16,48 = tx15,41 = mx14,46
(010) x21,87
(001) h20d
20x11,61 = h2
0d0x15,68 = h20e0x15,65 = h2
0ix12,64 = h20jx12,60 =
h20kx12,55 = h2
0B4B21 = h0h2d0x15,65 = h0h2ix12,60 =h0h2jx12,55 = h0h2Pe0x11,61 = h0h2B4Q1 = h0h6d0P
2j =h0h6e0P
2i = h0h6iP2e0 = h0h6jP
2d0 = h0h6Pd0Pj =h0e0Ph2x11,61 = h0f0Px12,60 = h0Ph2x15,74 = h0B2x13,35 =h0x
′B5 = h0x′PD2 = h0R1B23 = h1d
20x12,55 = h1e0Px12,60 =
h1Pd0x12,64 = h1Pe0x12,60 = h1x′B23 = h1Q1 x10,27 =
h1Q1 x10,28 = h1B221 = h2
2ix12,55 = h22Pd0x11,61 = h2
2B4x′ =
h2h6d0P2i = h2h6iP
2d0 = h2d0Ph2x11,61 = h2Ph2x15,68 =h2B2P
2D1 = h2R1x10,27 = h2R1x10,28 = h5iR2 = h6d0iP2h2 =
h6f0P3d0 = h6Ph2iPd0 = h6kP 3h2 = d0Ph1x12,64 = d0B1B21 =
e0Ph1x12,60 = e0B1Q1 = e0B2R1 = f0Ph2x12,55 = f0B2x′ =
gPh1x12,55 = gB1x′ = Ph2B2B4 = iB2
2
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 197
Stem 1208 (1) h6Q2
9 (1) x9,118
10 (100) x10,120
(001) h0x9,118
(011) h1x9,117 = h4x9,99
11 (100) x11,113
(010) h0x10,120 = h2x10,114
(001) h20x9,118 = h2x10,113
12 (10000) x12,106
(00011) h20x10,120 = h0h2x10,114 = h1h6Q1 = h1x11,109 = h2h6R1
(01000) x12,107
(00100) h0x11,113
(00001) h30x9,118 = h0h2x10,113 = h2
1x10,116
13 (1) h0x12,106
14 (10) x14,110
(01) h20x12,106 = h2
1x12,100 = h3x13,93
16 (10) h23x14,79 = D2x10,27
(11) h4x15,68
17 (100) P 2x9,97
(110) ix10,76
(001) h0h23x14,79 = h0h4x15,68 = h0D2x10,27 = h2
2x15,90 = gx13,73 =x7,40R1
18 (100) rx12,55 = B24
(110) d0x14,79
(001) h0ix10,76 = h0P2x9,97
(101) kx11,61
19 (100) h0rx12,55 = h0B24 = h3P
2x10,76 = x9,40x′
(110) h0d0x14,79
(001) h20ix10,76 = h2
0P2x9,97
(101) h0kx11,61 = h1Pd0x10,76 = h2jx11,61 = h5x18,50 = h6iP j =c0Px12,80 = d0Ph1x10,76 = f0x15,65 = Pc0x12,80
(011) h2Px14,84 = Ph2x14,84
20 (100) P 2x12,80
(001) h30ix10,76 = h3
0P2x9,97 = h2
0rx12,55 = h20B
24 = h0h3P
2x10,76 =h0x9,40x
′ = h3ix12,55 = h3B4x′ = g2x16,32 = xx15,41
(011) h20d0x14,79 = h0h2Px14,84 = h0Ph2x14,84 = h1h4x18,68 =
h1d0x15,68 = h22x18,87 = h2
2Px14,79 = h2Ph2x14,79 = g2x12,44 =d1x16,48 = Ph1x15,81 = Ph1x15,82 = tx14,46 = R1B5 = Q1 x10,32
198 ROBERT R. BRUNER
Stem 1217 (10) x7,101
(01) h6D2
8 (1) h0h6D2 = h0x7,101
9 (10) x9,119
(01) h20h6D2 = h2
0x7,101 = h1h6Q2
10 (1) h0x9,119 = h1x9,118 = h2x9,116
11 (1) x11,116
12 (10) h1x11,113
(01) h0x11,116
13 (1) h20x11,116 = h1x12,106 = h2x12,100 = h3x12,93
14 (1) Px10,109
15 (10) h4x14,79
(01) h1x14,110
16 (100) d0x12,85 = e0x12,80
(010) x16,109
(001) h0h4x14,79
17 (1011) rx11,61
(0010) h0x16,109
(0001) h20h4x14,79 = h1h4x15,68 = Q2x10,32
(0101) P 2x9,99
18 (100) g3A′ = g3A = gmx7,40 = r2A′ = r2A = wx9,51
(010) h0P2x9,99 = h1P
2x9,97 = h5x17,52 = h6P2v = Ph1x13,91 =
P 2h1x9,97
(001) h20x16,109
(011) h0rx11,61 = G21 x′ = B4X1
19 (1) h30x16,109 = h2
0rx11,61 = h0G21 x′ = h0B4X1 = h3ix11,61 =h3x18,87 = x9,40R1
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 199
Stem 1228 (1) h1x7,101 = D2
3
11 (10) h6B21
(01) x11,118
12 (100) A2
(010) h0h6B21 = h2h6Q1
(001) h0x11,118 = D2A′′
(101) A′2 = A′A
(011) h2x11,109
(111) h5x11,61
13 (100) x13,113
(010) h21x11,113 = h4x12,85 = D3X1 = x6,53B1
(001) h20h6B21 = h0h2h6Q1 = h0h2x11,109 = h0h5x11,61 = h2
2h6x′ =
h6Ph2B2 = e0x9,99
14 (1) h0x13,113 = h3D2Q2
15 (10) x15,108
(01) x15,109
16 (100) nx11,61 = rx10,65
(010) x16,112
(001) h0x15,109 = h21x14,110
17 (10) g3H1 = gtx7,40 = nrA′ = nrA = H1r2 = mx10,60 = Q2B23 =
Nx9,51 = x8,33B4 = x8,57w
(01) h0x16,112
18 (100) Pe0x10,76
(010) h1P2x9,99 = Ph1Px9,99 = qx12,55 = PD3x
′ = P 2h1x9,99
(110) d0x14,82
(001) h20x16,112 = h3x17,94 = xx13,46 = rx12,58 = G21 R1
(011) X21
200 ROBERT R. BRUNER
Stem 1238 (1) x8,113
9 (10) x9,121
(01) h0x8,113
10 (100) x10,124
(110) h6B4
(001) h0x9,121 = h2x9,118
11 (10000) x11,119
(01000) x11,120
(00001) h20x9,121 = h0h2x9,118
(01100) h5x10,65 = H1A′ = H1A
(00010) h0h6B4 = h2x10,120
12 (100) x12,116
(010) h0x11,119
(001) h20h6B4 = h0h2x10,120 = h1h6B21 = h2
2x10,114 = h2x11,113 =h6d0B1 = Ph2x7,97
13 (100) x13,116
(010) x13,117
(001) h20x11,119 = h2x12,106 = h3D
22 = h5x12,58 = r1Q2
14 (100) Q2x7,40
(001) h30x11,119 = h0h2x12,106 = h0h3D
22 = h0h5x12,58 = h0r1Q2 =
h1x13,113 = h23x12,86 = h3x13,97 = px10,63 = D3x10,27 = H1X1 =
qx8,75 = Ax8,32
(101) rx8,78 = A′x8,33 = Ax8,33
(011) h0x13,116 = h0x13,117
15 (100) x15,110
(010) h4x14,82 = d1x11,61 = nx10,65 = Q3Q1
(001) h20x13,116 = h2
0x13,117
16 (010) h0x15,110
(110) h1x15,109 = h4x15,74 = c0x13,95 = d0x12,86
(001) h30x13,116 = h3
0x13,117 = h3x15,96
(011) h23y14,83 = A′x10,27
17 (1101) h3x16,95
(0011) h0h23y14,83 = h0A
′x10,27 = h1x16,112 = e1x13,46 = g2x13,42 =nx12,58 = xx12,48 = qx11,61 = Q2B5 = Q2PD2 = x8,32X1
(0100) d0x13,87
(0010) h20x15,110
(0001) h40x13,116 = h4
0x13,117 = h0h3x15,96 = ix10,82
(0101) jx10,76
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 201
Stem 1246 (1) x6,94
7 (100) x7,103
(010) h6A
(001) h0x6,94
(011) h6A′
8 (1) x8,114
9 (100) x9,123
(010) x9,124
(001) h0x8,114
10 (00110) H21
(10000) x10,127
(01000) x10,128
(00100) h1x9,121
(00001) h20x8,114
(00010) h0x9,123 = h2x9,119
(00101) h6X1
11 (10000) x11,124
(00011) h3x10,113
(01000) x11,125
(00100) x11,126
(00001) h30x8,114 = h0h6X1 = h3x10,114
(00010) h0x10,127
12 (00110) h0x11,125
(10000) h1x11,120
(01000) h0x11,124
(00011) h0h3x10,113
(00100) h0x11,126 = h2x11,116 = px8,75 = n1Q2 = D2r1
(00001) h40x8,114 = h2
0h6X1 = h0h3x10,114 = h3h6R1
(10100) Q3Q2
(00010) h20x10,127 = h1x11,119 = h3x11,103
13 (1000) h5x12,60 = e0x9,102 = gx9,97 = nx8,78 = H1x8,33 = D2x7,40 =A′x7,34 = Ax7,34
(1100) h4x12,86
(1111) H1x8,32
(0001) h50x8,114 = h3
0h6X1 = h20h3x10,113 = h2
0h3x10,114 = h20x11,124 =
h0h3h6R1 = h3x12,96
continued
202 ROBERT R. BRUNER
Stem 124 continued14 (10000) x14,117
(00011) h1x13,116
(01000) x14,118
(00001) h1x13,117 = h4x13,87 = d0x10,102
(10100) rx8,80
(00010) d1x10,65
15 (0011) h0x14,118
(1000) x15,113
(0100) x15,114
(0110) h4x14,84
(0001) h0x14,117
(0101) h23x13,88 = h4y14,83 = H1x10,27
16 (00110) h0x15,113
(10000) x16,117
(00011) h20x14,118 = h0h4x14,84 = h2h4x14,79 = D2PD2
(01000) h1x15,110 = h3x15,98 = d1x12,58 = e1x12,48 = g2x12,44 = qx10,65 =x2
8,32
(10011) e0x12,85 = gx12,80
(00001) h20x14,117 = h0x15,114
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 203
Stem 1255 (1) x5,77
6 (10) h6H1
(01) h0x5,77
7 (1) h20x5,77 = h1x6,94
8 (10) x8,115
(01) x8,116
9 (01101) h6x8,32
(10000) x9,126
(01000) h5x8,80
(00001) h0x8,116
(00010) x9,129
(00101) h6x8,33
10 (10000) x10,132
(01000) x10,133
(00001) h20x8,116
(00111) h0x9,126
(00010) h0x9,129 = h3x9,116
11 (01101) h1x10,127
(00011) h0x10,132
(11000) h6x10,27 = h6x10,28
(01011) Q3D2
(00001) h30x8,116 = h2
0x9,126 = h0x10,133
(00010) h20x9,129 = h0h3x9,116
(01111) h3x10,116
(00101) h1x10,128 = h4x10,102
12 (10000) x12,124
(00011) h30x9,129 = h2
0h3x9,116 = h0h3x10,116
(01000) x12,125
(00100) h0h6x10,27 = h0h6x10,28 = h2h6B21 = h6d0B2
(00001) h40x8,116 = h3
0x9,126 = h20x10,133
(00010) h20x10,132 = h0Q3D2 = h2x11,118 = AA′′
continued
204 ROBERT R. BRUNER
Stem 125 continued13 (00011) h4
0x9,129 = h30h3x9,116 = h2
0h3x10,116 = h22x11,109 = h2h5x11,61 =
gx9,99
(00100) h0x12,124
(00001) h50x8,116 = h4
0x9,126 = h30x10,133 = h2
0h6x10,27 = h20h6x10,28 =
h0h2h6B21 = h0h6d0B2 = h22h6Q1 = c0x10,114 = d0x9,109
(01001) nx8,80
(00010) h30x10,132 = h2
0Q3D2 = h0h2x11,118 = h0AA′′ = h0x12,125 =
h21x11,120 = h1Q3Q2 = h2D2A
′′ = h2A′2 = h2A
′A = f0x9,102 =rx7,79
(11001) h3x12,100
14 (001) h20x12,124 = h0h3x12,100 = h3A
′Q2 = px10,65
(011) h1h4x12,86 = c0x11,103 = D3x10,32
(111) Px10,113
15 (00110) h1x14,118
(10000) x15,117
(11000) Q2G21
(00100) x15,119
(00001) h30x12,124 = h2
0h3x12,100 = h0h3A′Q2 = h0px10,65 = h2
1x13,116 =h1d1x10,65 = h3x14,104 = D2x9,40 = A′′X1 = x7,33x8,32
(00010) h0Px10,113
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 205
Stem 1262 (1) h2
6
3 (1) h0h26
4 (10) D31
(01) h20h
26
5 (1) h30h
26
6 (10) x6,97
(01) h40h
26
7 (01) h50h
26
(11) h1h6H1
8 (100000) x8,117
(010000) x8,118
(001000) x8,119
(001110) h6x7,34
(000100) x8,120
(000001) h60h
26
9 (100000) x9,131
(010000) h1x8,115
(000011) h0x8,118
(001000) h1x8,116
(010011) h2x8,113
(000100) h0x8,117
(000001) h70h
26
10 (10000) x10,137
(00011) h20x8,118 = h0h2x8,113 = h1x9,126 = h1x9,129 = d2D2
(00001) h80h
26
(01001) h0x9,131
(00101) h20x8,117 = h1h5x8,80 = h1h6x8,32 = h3x9,117
(01010) h2x9,121
11 (100000) x11,134
(000011) h20x9,131 = h0h2x9,121 = h2
2x9,118
(001000) h1x10,133
(011000) h1x10,132 = h3x10,118
(000100) h0x10,137 = h2h6B4 = d0x7,97
(000001) h90h
26
continued
206 ROBERT R. BRUNER
Stem 126 continued12 (00110) h0x11,134 = h2
1x10,127 = h21x10,128 = h1h3x10,116 = h1h4x10,102 =
h2h5x10,65 = h2H1A′ = h2H1A = c0x9,115 = c1x9,102 = nx7,79
(00011) h20x10,137 = h0h2h6B4 = h0d0x7,97 = h1h6x10,27 = h1h6x10,28 =
h22x10,120 = h2x11,119 = h6e0B1 = c0x9,116 = d0x8,105
(10011) d1x8,80
(01011) h3x11,109
(00001) h100 h2
6
13 (100) h1x12,125
(010) h0h3x11,109 = h3H1Q2 = h3D2A′
(110) h1x12,124
(001) h110 h2
6
14 (0100) x14,126
(0010) h1h3x12,100 = h2x13,116 = qx8,80
(1010) D2G21
(0001) h120 h2
6
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 207
Stem 1271 (1) h7
2 (1) h0h7
3 (10) h1h26
(01) h20h7
4 (1) h30h7
5 (1) h40h7
6 (10) x6,99
(01) h50h7
7 (00110) h2x6,94
(10000) x7,109
(01000) x7,110
(00100) h1x6,97
(00001) h60h7
(00010) h0x6,99 = h6A′′
8 (0011010) h2h6A′
(0011000) h2h6A
(0001100) h0x7,109
(0100000) x8,124
(0000100) h0x7,110
(0000001) h70h7
(1001000) h2x7,103 = h4x7,97
(0000010) h20x6,99 = h0h2x6,94 = h0h6A
′′
9 (10000) h1x8,118 = h1x8,120
(00100) h0x8,124
(00001) h80h7
(11100) h1x8,119
(00010) h20x7,109 = h1x8,117
(10010) h3h6Q2 = h5x8,83
(01010) h2x8,114
10 (010000) x10,143
(000010) h20x8,124 = h0h2x8,114 = h1x9,131
(101000) h2x9,123
(001110) h3x9,118
(000100) h21x8,116
(000001) h90h7
continued
208 ROBERT R. BRUNER
Stem 127 continued11 (0100) h0x10,143
(0010) h0h3x9,118
(1010) h0h2x9,123 = h1x10,137 = h22x9,119 = h2H
21 = h2x10,127 =
h4x10,107 = d1x7,79 = D3x7,40
(0001) h100 h7
12 (100) h1x11,134 = h24x10,76 = c0x9,117 = Q3B3
(010) h20x10,143 = h2x11,124 = px8,80 = A′r1
(110) h3x11,113
(001) h110 h7
13 (00110) gx9,102
(10110) h3x12,106
(01000) x13,132
(00100) h5x12,64 = A′x7,40 = Ax7,40
(00001) h120 h7
(00010) h30x10,143 = h0h2x11,124 = h0h3x11,113 = h0px8,80 = h0A
′r1 =h1d1x8,80 = nx8,83 = rx7,81
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 209
Stem 1282 (1) h1h7
4 (1) h21h
26
5 (1) x5,80
6 (1000) x6,101
(0100) x6,102
(0010) h2x5,77
(0001) h0x5,80
7 (010) h0x6,101
(110) h2h6H1
(001) h20x5,80 = h0h2x5,77 = h0x6,102 = h1x6,99
8 (1) h3h6D2 = h3x7,101 = h5x7,81
9 (100) h2x8,115
(010) h1x8,124 = D3Q3
(001) h0h3h6D2 = h0h3x7,101 = h0h5x7,81
10 (0011) h2h6x8,33 = h2x9,126
(1000) x10,148
(0100) x10,149
(0010) h2h6x8,32 = h4x9,111
(0001) h20h3h6D2 = h2
0h3x7,101 = h20h5x7,81 = h2
1x8,118 = h21x8,119 =
h21x8,120 = h1h3h6Q2 = h1h5x8,83 = h2h5x8,80
11 (100) h5x10,76 = h6B23
(010) h0x10,148 = h2x10,132 = px7,79 = H1r1 = n1A′ = Q3A
′
(001) h0x10,149 = h1x10,143
(011) h4x10,109 = Q3A
12 (10000) x12,137
(00100) h3x11,116
(00001) h0h5x10,76 = h0h6B23 = h2h6x10,27 = h2h6x10,28 = h6e0B2 =c0x9,118
(11001) H1x7,40
(00010) x12,140
210 ROBERT R. BRUNER
Stem 1293 (10) h2h
26
(01) h21h7
4 (1) h0h2h26
5 (100) H11
(010) h2D31
(001) h20h2h
26 = h3
1h26
7 (1) h2x6,97
8 (10) h6x7,40
(01) x8,132
9 (0011) h0h6x7,40 = h2h6x7,34 = h2x8,119
(1000) x9,145
(0100) x9,146
(0001) h0x8,132 = h2x8,118
10 (100) x10,152
(010) h0x9,146 = h2x9,131
(001) h20x8,132 = h0h2x8,118 = h2
1x8,124 = h1D3Q3 = h22x8,113 =
h4x9,112 = c0x7,101 = d2A
11 (100000) x11,147
(010000) x11,148
(000011) h0x10,152 = h2x10,137 = e0x7,97
(001000) h6B5 = h6PD2
(000100) h1x10,149
(000001) h20x9,146 = h0h2x9,131 = h2
2x9,121
Stem 1302 (1) h2h7
3 (1) h0h2h7
4 (1) h20h2h7 = h3
1h7
6 (1) h6n1
7 (1) x7,118
8 (10) x8,133
(01) h2x7,109
9 (100) h3x8,113
(010) h1h6x7,40 = h22h6A
′ = h22h6A
(001) h0x8,133 = h2x8,124
10 (10000) x10,155
(00100) h3x9,121
(00001) h20x8,133 = h0h2x8,124 = h1x9,146 = h2
2x8,114
(01100) h5x9,86 = h6x9,39
(00010) h0h3x8,113
THE COHOMOLOGY OF THE MOD 2 STEENROD ALGEBRA 211
Stem 1317 (10) h2x6,101
(01) h3x6,94
8 (0011) h0h2x6,101 = h1x7,118 = h3x7,103 = D3p1
(1000) h22h6H1 = h3h6A
(0100) x8,136
(0001) h0h3x6,94
(1001) h3h6A′
9 (00011) h1x8,133
(00100) x9,154
(00001) h0x8,136
(01001) h6G21
(10010) h3x8,114
Stem 1324 (1) h2
2h26
6 (100) h3x5,77
(010) x6,107
(001) h2H11
7 (010) h0h3x5,77
(001) h0x6,107
(101) h3h6H1
8 (10000) x8,139
(01000) x8,140
(00100) h22x6,97
(00001) h20x6,107
(00010) h20h3x5,77 = h1h3x6,94 = h5x7,88
Stem 1333 (10) h3h
26
(01) h22h7
4 (1) h0h3h26
5 (01) h20h3h
26
(11) h3D31
6 (1) h30h3h
26
7 (100) x7,124
(110) h3x6,97
(001) h1x6,107
Stem 1342 (1) h3h7
3 (1) h0h3h7
4 (10) h1h3h26
(01) h20h3h7
5 (10) h26c0
(01) h30h3h7
6 (1) x6,110
212 ROBERT R. BRUNER
Stem 1353 (1) h1h3h7
4 (1) h7c0
5 (1) h21h3h
26 = h3
2h26
Stem 1364 (1) h2
1h3h7 = h32h7
Stem 137
Stem 138
Stem 139
Stem 140
References
1. Robert R. Bruner, “Calculation of large Ext modules”, Computers in Geometry and
Topology (M. C. Tangora, ed.), Marcel Dekker, New York, 1989, 79-104.2. R.R.Bruner, “Ext in the nineties”, pp. 71-90 in Algebraic Topology, Oaxtepec 1991,
Contemp. Math. 146, Amer. Math. Soc., Providence, 1993.3. Robert R. Bruner, “Root invariants and Sq0’s in ExtA’, preprint, April 1997.
Mathematics Department, Wayne State University, Detroit, Michigan, 48202
E-mail address: [email protected]
Top Related