Years Ago Allen Shields*

Introduction. This depar tment of the Intelligencer was previously known as "50 and 100 Years Ago." The change in name indicates a change in policy: we shall endeavor to present interesting mathematics from the past that is approximately associated to some anniversary (e.g., 50, 75, 100 years), without being tied rigidly to specific years. We will of course wel- come suggest ions, comments , and criticism from readers�9

H o m o t o p y t h e o r y . G.W. Whitehead, in the preface to his book on homotopy theory (Whitehead [1978]), states: "Except for the fundamental group, the subject of homotopy theory had its inception in the work of L. E. J. Brouwer . . . . " The paper in ques t ion is Brouwer [1912]. Thus we are at the 75 th anniversary of this fundamental work. (The same issue of the Mathe- matische Annalen contains his extension of the Jordan curve theorem to Euclidean n-space.) Just over 50 years ago there was another major advance in ho- motopy theory: the four notes of Hurewicz ([19351,2], [19361,2] ). In these notes he defined the higher ho- motopy groups and gave a number of their properties. In part icular , in the second note he stated, and ske tched a proof of, the Hurewicz i somorph i sm theorem; as stated by him this says: "If the first n - 1 (n /> 2) homotopy groups of a connected, locally con- tractible space vanish (that is, reduce to the identity element), then the r/th homotopy group is isomorphic to the n th homology group." As regards the definition of the higher homotopy groups, in footnote 2 he states that after the appearance of the first note he learned that "a definition of the homotopy groups equivalent to mine (though in an early different form) was given by Cech in an address to the International Congress of Ma thema t i c i ans in Z~irich in 1932 (see the Pro- ceedings, vol. II, p. 194). Herr Cech has kindly written to inform me that actually the homotopy groups were known earlier to Dehn, though they do not appear in any of his publications."

Cohomology theory. It was about 50 years ago that cohomology theory was discovered, though as Eilen-

* C o l u m n editor 's address: Mathematics Department, University of

Michigan, Ann Arbor, MI 48109, USA

berg and Steenrod [1952] (p. 48) state: "The origin of the cohomology groups is thoroughly confused." The "co" terminology seems to be due to Whitney [1938], who pointed out that the "pseudocycles" of Lefschetz, and the cycles in the "dual complex" considered by Mayer and then by Tucker, were really cocycles (see Whitney for these references). A decisive step was taken with the introduction of products; the resulting ring structure for cohomology was richer than the group structure for homology. Products were discov- ered independently and simultaneously by Alexander and by Kolmogorov and were presented by these two authors at the First International Topology Confer- ence, Moscow, 4-10 September 1935. This conference was a landmark and was attended by most leading to- pologists (see Matem. Sbornik 1143], 1936, p. 793 for the list of participants).

Alexander [1936] indicates that he was led to intro- duce cohomology groups by a consideration of cycles with coefficients from the reals modulo 1, introduced by Pontrjagin. The k th homology group, with these co- efficients, of a compact metric space is an abelian com- pact metric group. Pontrjagin [1934] had shown that such a group is the character group of a countable dis- crete group. Alexander continues: "This immediately suggests the advisability of regarding the discrete group . . . as the k th invariant of the space, and of looking for a revised theoretical treatment leading simply and directly to this group . . . . One decided advantage of taking the discrete g r o u p s . . , is that we can then define the product (as distinguished from the sum) of two elements of the same or of different groups . . . . Essentially the same theory was also de- ve loped , qui te i n d e p e n d e n t l y , by A. Kolmogo- rov . . . . The definition of the p r o d u c t . . , as originally given by Kolmogorov and the present author has r e -

c e n t l y been modified by ~ech [1936], so that the r e -

v i s e d product is a pth part of the original one, where p is a constant depending on the dimensions of the two factors. This modification is an essential improvement �9 . . nor is it in any way trivial . . . . We have adopted Cech's definition in this paper, although the change has necess i ta ted a complete recast ing of all our proofs." In footnote 7 Alexander states that a modifi- cation similar to that of Cech was suggested indepen-

6 THE MATHEMATICAL INTELLlGENCER VOL. 9, NO. I �9 1987 Springer-Verlag New York

dent ly by Whi tney in a letter to L. Zippin. Reading Kolmogorov [19362], one sees that he too

was inf luenced by Pontr jagin 's theory. In this paper he int roduces cohomology groups , but not products . He states that he obtained the results of the paper (for the case of finite simplicial complexes) in the spring and s u m m e r of 1934, and that they were presen ted at an i n t e r n a t i o n a l c o n f e r e n c e on t e n s o r a n a l y s i s (Moscow) in May 1934. His theory of products appears in Kolmogorov [19363]. Curiously, Kolmogorov 's role in cohomology theory is somet imes forgotten; for ex- ample, Eilenberg and Steenrod [1952] p. 48 state: "The first intrinsic definit ion of the cohomology groups was g iven by A lexande r at the M o s c o w Confe r en ce in 1936." And Spanier [1966] chap. 6, w p. 306, entitles the section "The Alexander cohomology theory . " The second sentence of the text refers to the Alexander- Kolmogorov construct ion but this seems to be the only reference to Kolmogorov ( though not to Alexander).

Exact Sequences . We conc lude with a few words a b o u t exact s equences , wh ich were d i s cove red by Hurewicz a little less than 50 years ago. Unfor tunate ly he on ly p u b l i s h e d a br ief abs t rac t (see H u r e w i c z [1941]) whose text we reproduce below. Surprisingly, the concept seems to have been neglected for several years. Eilenberg and Steenrod [1945] take the exact- ness of the homology sequence of a pair of spaces as Axiom 5 in their axioms for homology theory. An indi- cation of h o w little exactness had penet ra ted into the mathemat ical consciousness is given by their next sen- tence. "At first glance, this axiom may seem strange e v e n to one famil iar w i th h o m o l o g y t h e o r y . " The "Th6or~me fondamenta l " (Thm. 5.1) of Cartan [1945] states that the p roduc t of two consecutive homomor - phisms in the homology sequence of a pair is zero. (He adds tha t if the coeff ic ient g roup is the circle g roup T, then the kernel of each h o m o m o r p h i s m coin- cides exactly with the range of the preceding homo- morph i sm. ) Nei ther of these papers refers to Hure- wicz. Nei ther Hurewicz nor these authors used the te rm "exact . " This term is used by Kelley and Pitcher [1947], w h o gave wha t was probably the first system- atic t rea tment of exact sequences in homology and co- h o m o l o g y t h e o r y (see E i lenberg ' s r ev i ew of the i r paper , MR 9, p. 52), and Ei lenberg-Steenrod [1952] p. 47.

Witold Hurewicz: On duality theorems.

Let A be a locally compact space, B a closed subset of A, and Hn(A), Hn(B), H n ( A - B ) the n-dimensional cohomology groups of the sets A, B and A - B (with integers as coeff ic ients) . Cons ide r " n a t u r a l h o m o m o r p h i s m s " Hn(A) ~ Hn(B) ~ H~+I(A-B) ~ H"+I(A) ~ H~+I(A-B). It can be shown that the kernel of each of these homomor- phisms is the image of the preceding homomorphism. This statement contains Kolmogoroff's generalization of Alex-

ander's duality theorem and has many applications. Using the preceding theorem one can prove it: If A and B are com- pact spaces of dimensions n and m respectively, the neces- sary and sufficient condition that the topological product A x B be of dimension n + m is the existence of an open Set U C A and an open set V C B such that Hn(U) and Hm(V) con- tain elements c~ and [3 satisfying the following conditions: If the integer d is a factor of the order of R, then [B # 0 modulo d (that is, there is no element ~ of Hm(V) satisfying [3 = d'y); if the integer e is a factor of the order of [3, then c~ # 0 modulo e. (Received May 3, 1941.)

Bibliography

J. W. Alexander [19351], "On the chains of a complex and their duals," Proc. Nat. Acad. Sci. USA 21, 509-511.

- - [ 1 9 3 5 2 ] , "On the ring of a compact metric space," ibid. 21, 511-512.

- - [ 1 9 3 6 ] , "On the connectivity ring of an abstract space," Annals of Math.. 37, 698-708.

L. E. J. Brouwer [1912], "Uber Abbildungen von Mannigfal- tigkeiten," Math. Ann. 71, 97-115; see also p. 598.

Henri Cartan [1945], "M6thodes modernes en topologie al- g6br!que," Comm. Math. Helvet. 18, 1-15.

Eduard Cech [1936], "Multiplications on a complex," Annals of Math. 37, 681-697.

Samuel Eilenberg and Norman E. Steenrod [1945], "Axiom- atic approach to homology theory," Proc. Nat. Acad. Sci. USA 31, 117-120.

- - [ 1 9 5 2 ] , Foundations of algebraic topology, Princeton Uni- versity Press.

Witold Hurewicz [19351], "Beitr/ige zur Topologie der Defor- ma t ionen (I. H 6 h e r d i m e n s i o n a l e n Homotop ie - gruppen)," K. Akad. Weten. (Amsterdam), Proc. Sect. Sci. 38, 112-119.

- - [ 1 9 3 5 2 ] , "Beitr/ige zur Topologie der Deformationen (II. Homotopie- und Homologiegruppen)," ibid. 38, 521-528.

- - [ 1 9 3 6 1 ] , "Beitr~ige zur Topologie der Deformationen (III. Klassen und Homologietypen von Abbildungen)," ibid. 39, 117-126.

- - [ 1 9 3 6 2 ] , "Beitr/ige zur Topologie der Deformationen (IV. Asparische R/iume)," ibid. 39, 215-224.

- - [ 1 9 4 1 ] , "On duality theorems," Bull. Amer. Math. Soc. 47, 562-563.

John L. Kelley and Everett Pitcher [1947], "Exact homomor- phism sequences in homology theory," Annals of Math. 48, 682-709; Math. Rev. 9, p. 52.

Andrei N. Kolmogorov [19361], "Cycles relatifs. Th6or~me de dualit6 de M. Alexander," C. R. Acad. Sci. (Paris) 202, 1641-1643.

- - [ 1 9 3 6 2 ] , "Uber die Dualit/it im Aufbau der kombina- torischen Topologie," Matem. Sbornik 1(43), 97-102.

- - [ 1 9 3 6 3 ] , "Homologierung des Komplexes und des lokal-bikompakten Raumes," ibid. 1(43), 701-706.

Lev S. Pontrjagin [1934], "The theory of topological commu- tative groups," Annals of Math. 35 (1934), 361-388.

Edwin H. Spanier [1966], Algebraic topology, McGraw-Hill, New York.

George W. Whitehead [1978], Elements of homotopy theory, Springer-Verlag (Grad. Texts Math. 61), New York, Berlin, Heidelberg.

Hassler Whitney [1937], "On products in a complex," Proc. Nat. Acad. Sci. USA 23, 285-291.

- - [ 1 9 3 8 ] , "On products in a complex," Annals of Math. 39, 397-432.

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