The E (1; 2) Cohomology of the

Eilenberg-Mac Lane Space K (Z; 3)

Hsin-hao Su

March 21, 2006

1 Eilenberg-Mac Lane Spaces

Let X be a CW-complex and X(n) be its n-th skeleton. We have the followingdiagram

X(0) ,! X(1) ,! � � � ,! X(n�1) ,! X(n) ,! � � � ,! X# #W

�1Sn�1

W�nSn�n

where Sn is the n-sphere.

The suitable analogue for homotopy is a space whose homotopy is concentratedin one dimension.

De�nition 1.1 Suppose that G is an abelian group and n � 0 or G is a groupand n = 0 or 1. If X is a space such that

�j (X) =

(G, if j = n,f0g , if j 6= n,

where �j (X) is the j-th homotopy group, [Sn; X], all the equivalent classesof continuous maps up to homotopy, then X is called an Eilenberg Mac-Lanespace of type (G;n).

Theorem 1.2 If G is an abelian group and n � 0 or G is a group and n = 0or 1, then there exists an Eilenberg-Mac Lane space of type (G;n).

Theorem 1.3 If two spaces are Eilenberg-Mac Lane space of type (G;n),then they have the same homotopy type. For any such space, we write it asK (G;n).

Example 1.4 S2 ' K (Z; 1).

Example 1.5 RP1 ' K (Z=2; 1).

Example 1.6 CP1 ' K (Z; 2).

If given a K (G;n), the path-loop �bration

K (G;n) �! PK (G;n)#

K (G;n)

gives a long exact sequence on homotopy that implies that K (G;n) �=K (G;n� 1).

For a abelian group G, there is a natural isomorphism, for each n and eachCW-complex Y , Hn (Y;G) �= [Y;K (G;n)].

2 Generalized Cohomology Theory

In 1962, E. Brown proved that for any generalized cohomology theory E, forall k, there exists a classifying space Ek such that

Ek (X) �= [X;Ek] ,

the set of homotopy classes of unbased maps X �! Ek.

De�nition 2.1 A spectrum is a sequence of spaces, Ek, with basepoint, to-gether with weak equivalences "k : �Ek ! Ek+1, or adjointly, "

0k : Ek !

Ek+1.

De�nition 2.2 The generalized cohomology theory associated to fEkg, is de-noted by E� (X) and de�ned by Ek (X) = [X;Ek] for a space X.

3 BP and E (n; k) Spectra

There is the Brown-Peterson cohomology, BP � (�), associated with a primep. The coe¢ cient ring for Brown-Peterson cohomology is

BP � �= Z(p) [v1; v2; � � � ]

where the degree of vn is �2 (pn � 1). This Brown-Peterson theory can belabeled as P (0).

Next, there are theories introduced by Morava, P (n) with coe¢ cient ringBP ��In where In = (p; v1; � � � ; vn�1) which can be constructed by usingBaas-Sullivan singularities. Thus, they come equipped with stable �brations

�2(pn�1)P (n) vn�! P (n) �! P (n+ 1) ,

which gives us long exact sequences in cohomology.

Similarly, for all q, we have theories BP hqi with coe¢ cient rings

BP hqi� �= Z(p) [v1; v2; � � � ; vq]equipped with a stable co�bre sequence

�2(pq�1)BP hqi

vq�! BP hqi �! BP hq � 1i �! �2pq�1BP hqi .

Using Baas-Sullivan singularities and localization, we can construct theoriesE (k; n) with coe¢ cient rings

E (k; n)� �= v�1n BP hni��Ikfor all 1 � k � n. Also, we have similar stable co�brations

�2�pk�1

�E (k; n)

vk�! E (k; n) �! E (k + 1; n)

and long exact sequences. In particular, for 1 � k � n,

E (k; n)� �= v�1n BP (n)��Ik = Fphvk; vk+1; � � � ; vn; v�1n

i.

A special case, when k = n > 0, is the nth Morava K-theory, K (n)� (X),with

K (n)� �= Z=phvn; v

�1n

i.

Also, when k = 0,

E (n)� = E (0; n)� �= v�1n BP hni� = Z(p)hv1; v2; � � � ; vn; v�1n

i.

For BP , we have a BP -tower

BP �! � � � �! BP hni �! � � � �! BP h1i �! BP h0i �! K (Zp) .

Note that BP h0i is the Eilenberg-Mac Lane spectrum K (Z) for the ring Zand BP h1i is connective K-theory bu. Also, K (Zp) is the Eilenberg-MacLane spatrum for the ring Zp.

Let k � 2�pn + pn�1 + � � �+ p+ 1

�. In 1973, Wilson [Wil75] found the fol-

lowing homotopy equivalence, namely Wilson Splitting Theorem, among spacesof the -spectra of BP and BP hni:

BP �= BP hnik�

Yj�n+1

BP hjik+2(pj�1)

and if m � n, then

BP hmik�= BP hni

k�

mYj=n+1

BP hjik+2(pj�1)

.

Here in the case m = 1, n = 0 and k = 2, we have

BP h1i2�= BP h0i

2�BP h1i

4

where BP h0i2= K (Z; 2) = CP1.

4 History

In 1998, Ravenel, Wilson and Yagita [RWY] found that, for a prime p,

BP ��K�Z(p); q + 2

�� �= BP ��BP hqi

g(q)

��v�q� ,

where g (q) = 2�pq + pq�1 + � � �+ p+ 1

�and

�v�q�denotes the ideal gener-

ated by the image under v�q of the augmentation ideal inBP��BP hqi

g(q)�jvqj

�.

In 1999, Tamanoi [T] who was motivated by the above result, created newgenerators and wrote down a very simple relation of the fundamental class interms of his new generators.

We take p = 2 and q = 1.

By the short exact sequence of Hopf algebras

BP � � BP � (K (Z; 3)) f� � BP ��BP h1i

6

� v�1 � BP ��BP h1i

4

� � BP �,

we have

BP � (K (Z; 3)) �=BP �

�BP h1i

6

��v�1� ,

where�v�1�denotes the image under v�1 of the augmentation ideal ofBP

��BP h1i

4

�.

Tamanoi�s generator, the class �1+j with j � 1 comes from the followingcomposition:

BP h1i6�! BP 6

v1�! BP 4projh1+ji�! BP h1 + ji

2+2(1+j)+1�! BP

2+2(1+j)+1.

Note that �1+j 2 BP 2+2(1+j)+1 �

BP h1i6

�has the same degree as y(j).

The relation Tamanoi found in this case is

v�1�x(0)

�= v1y(0) + v2�2 + � � �+ vi�i + � � � .

By tensoring with E (1; 2)�, we have

E (1; 2)� (K (Z; 3)) �=E (1; 2)�

�BP h1i

6

��v�1�

where

v�1�x(0)

�= v1y(0) + v2�2.

5 Formal Group Laws

De�nition 5.1 A formal group law is a power series

F (y; z) =Xi;j�0

aijyizj

such that

1. F (y; 0) = y and F (0; z) = z,

2. F (y; F (z; w)) = F (F (y; z) ; w),

where aij is in a commutative ring with unit. Moreover, we say a formal grouplaw is commutative if F (y; z) = F (z; y). We write y +F z for F (y; z).

De�nition 5.2 In BP� (BP �) [[s; t]], we de�ne

b (s) +[F ] b (t) = b (s) +[F ]BPb (t) = �

i;j�0

haiji� b (s)�i � b (t)�j .

Proposition 5.3 (In [RW77] Theorem 3.8 on Page 256) In BP� (BP ) [[s; t]],we have relations

b ([2]F (s)) = [2][F ] (b (s)) .

Proposition 5.4 (In [RW77] Theorem 3.11(b) onPage 257) In BP for theprime 2, we have

[2]F (t) =XFBP

n>0vnt

2nmod (2) .

6 Generators

We need to compute v�1 : E��BP h1i

4

��! E�

�BP h1i

6

�. These are power

series rings. We compute on the indecomposables, Qv�1 : QE��BP h1i

4

��!

QE��BP h1i

6

�, which is dual to Pv1� : PE�

�BP h1i

6

��! PE�

�BP h1i

4

�.

The latter is simply multiplication by [v1].

The double suspension b(0) � � factors as

E��BP h1i

4

��! QE�

�BP h1i

4

� �=�! PE��BP h1i

6

�,! E�

�BP h1i

6

�and similarly for PE�

�BP h1i

4

�. We therefore compute the homomorphism

of free E�-modules

v1� : QE��BP h1i

4

��! QE�

�BP h1i

2

�.

The relationship between E�, QE� and PE� in a diagram:

E��BP h1i

4

� v1��! E��BP h1i

2

�# #

QE��BP h1i

4

� Qv1��! QE��BP h1i

2

��=# � � b(0) #

PE��BP h1i

6

� Pv1��! PE��BP h1i

4

�\ \

E��BP h1i

6

� v1��! E��BP h1i

4

� Dx(i);�

E�! E�

# # %QE�

�BP h1i

6

� Qv1��! QE��BP h1i

4

�

(1)

whereDx(i);�

Eis the inner product in the dual.

Theorem 6.1 The elements bi for i > 0 form a basis of the free E�-moduleQE�

�BP h1i

2

�.

We introduce the Bendersky generators hi of QE� (BP �). (see [B82]) Theymay be de�ned by

b (x) = h0x+[F ] h1x2 +[F ] h2x

4 +[F ] � � � (2)

where [F ] denotes the formal group law using the right BP�-action by theelements [vi].

Lemma 6.2 Given n > 0, write n in binary form as n = 2i0+2i1+ � � �+2ik,with i0 < i1 < � � � < ik. Then in QE� (BP 2),

bn �hvk1

ihi0hi1hi2 � � �hikmod I1 + ([v2] ; [v3] ; � � � ) :

To show that a set of elements is a basis, it is su¢ cient to work mod I1 =

(2; v1; v2; � � � ) and check that we have a basis over Fp, according to theNakayama Lemma. If we replace each b(i) by hi in the Ravenel-Wilson ba-sis of QE� (BP �), we obtain another basis.

Lemma 6.3 In QE��BP h1i

2

�, each bn can be expressed as a polynomial in

the elements b(i) and [v1], with no v1 or v2.

Remark 6.4 If n is not a power of 2, this polynomial is divisible by [v1].

Next, we consider QE��BP h1i

4

�.

Lemma 6.5 If n is not a power of 2, there is an element b̂n 2 QE��BP h1i

4

�such that [v1] b̂n = bn in QE�

�BP h1i

2

�.

In particular, by Lemma 6.2, b̂n �hvk�11

ihi0hi1hi2 � � �hikmod I1. The only

missing elements from the modi�ed Ravenel-Wilson basis of QE��BP h1i

4

�are the elements h2i � b2(i) for i � 0. We therefore take as a basis of

QE��BP h1i

4

�, the elements b̂n for n not a power of 2 and the elements

b2(i). The main relation, mod 2, isXqbq ([2] (x))

q = b ([2] (x)) � [v1] b (x)2 �Xi

[v1] b2ix2i.

Recall that ([2] (x))q =Xk�1

Cq;kx2k. Take coe¢ cients of x2

i+1. Then

(v1�b2(i) = [v1] b

2(i) =

Pq�2i Cq;2ibq, for i � 0;

v1�b̂n = [v1] b̂n = bn, for n not a power of 2(3)

gives v1� precisely in terms of our bases.

Let us consider the primitives.

By [RW77], a basis of the primitives PE��BP h1i

4

�comes from the basis of

QE��BP h1i

4

�by multiplying b(0).

Our basis of PE��BP h1i

4

�= PM4 � PR4 consists of two parts:

1. Elements b(0) � bn for n not a power of 2, which span the submodulePM4.

2. Elements b(0) � b(i) for i � 0, which span the submodule PR4.

Our basis of PE��BP h1i

6

�= PM6 � PR6 consists of two parts:

1. Elements b(0) � b̂n for n not a power of 2, which span the submodulePM6.

2. Elements b(0) � b�2(i) for i � 0, which span the submodule PR6.

In terms of these bases, v1� is given by8>><>>:v1�

�b(0) � b̂n

�= [v1] � b(0) � b̂n = b(0) � bn, for n not a power of 2

v1��b(0) � b�2(i)

�= [v1] � b(0) � b�2(i) =

Xq�2i

Cq;2ib(0) � bq.

(4)

Now, we can consider the duals QE��BP h1i

4

�and QE�

�BP h1i

6

�, the

indecomposables of the algebras E��BP h1i

4

�and E�

�BP h1i

6

�.

Proposition 6.6 (In [RWY] Page 6) Since the spaces BP hqikwhere k �

2�2q + 2q�1 + � � �+ 2 + 1

�are all torsion-free for ordinary homology, we

know that they are BP�-free and the Brown-Peterson cohomology is just theBP �-dual. Likewise, the induced cohomology homomorphisms are just thedual homomorphisms.

Proposition 6.7 (In [RWY] Section 8.5 at Page 39; Also in [RW77]) BothBP �

�BP h1i

6

�and BP �

�BP h1i

4

�are power series rings on generators

dual to the primitives in the BP -homology.

Dualizing to cohomology, we write E��BP h1i

4

�= A bM , where A and M

are power series rings generated by the duals of PR4 and PM4, respectively,and E�

�BP h1i

6

�= B bN , where B and N are power series rings gener-

ated by the duals of PR6 and PM6, respectively. Since v1� : PM6�= PM4

is an isomorphism, so is v�1 : M ! N (considered as quotient rings). Socoker

hv�1 : A bM ! B bNi �= coker

hv�1 : A! B

i. We therefore limit at-

tention to A and B. De�ne generators x(i) 2 A dual to b(0)�b(i) and y(i) 2 Bdual to b(0) � b�2(i), so that

A = E�hhx(i) : i � 0

iiand

B = E�hhy(i) : i � 0

ii.

We have a short exact sequence of (completed) Hopf algebras

E� � E� (K (Z; 3)) f� � E�hhy(i) : i � 0

ii v�1 � E�hhx(i) : i � 0

ii � E�.

ForE = E (1; 2), dualizing (4), in v�1 : QE��BP h1i

4

��! QE�

�BP h1i

6

�,

we have

v�1�x(i)

�=Xj�i

C2i;2jy(j).

Recall that v1t2 +F v2t4 =

Xk�1

Akt2k and

�v1t

2 +F v2t4�q=

Xk�1

Cq;kt2k.

Also, Cq;k = 0, if k < q and C2q;k = A2q

2�qk.for all q � 1 and k � 2q,interpreted as 0 if k is not divisible by 2q.

Thus, C2i;2j = A2i

2j�i = A2i

(j�i), where we write A(i) = A2i, and thatA(0) = A1 = v1 and A(1) = A2 = v2. Then, for i � 0, Qv�1 : QA �! QB

is given by

v�1�x(i)

�= v2

i

1 y(i) + v2i2 y(i+1) +

Xk�2

A2i

(k)y(i+k). (5)

Put z(i) = f��y(i)

�2 QE� (K (Z; 3)); these elements generate the E�-

module topologically, with respect to the z-�ltration in which Fn = FnQE� (K (Z; 3))consists of all elements

Pk�n

dkz(k), with dk 2 E�.

We also have the v1-�ltration, de�ned by the submodules�vk1

�= vk1QE

� (K (Z; 3)).Since v2 is invertible, we may rewrite the above relation as

z(i+1) = v2i1

0@v�2i2 z(i) +Xk�2

�A0(k)

�2iz(i+k)

1A .This shows that z(0) generates the module topologically, with respect to thev1 �ltration.

Proposition 6.8 The z-�ltration and the v1-�ltration de�ne the same topologyon QE� (K (Z; 3)).

We need to extend bases of PE��BP h1i

6

�and PE�

�BP h1i

4

�toE�

�BP h1i

6

�andE�

�BP h1i

4

�in order to de�ne x(i) and y(i) as elements ofE

��BP h1i

4

�and E�

�BP h1i

6

�.

We construct a basis ofE��BP h1i

6

�as follows. We have a basis of PE�

�BP h1i

6

�consisting of the elements b(0) � b�2(i) (for i � 0) and b(0) � b̂n (for n not apower of 2). We adjoin new basis elements b(k) � b�2(k+i) (for i � 0 and k > 0)and b(k) � b̂n;k (for n not a power of 2 and k > 0), where V kb̂n;k = b̂n.

This desired basis of E��BP h1i

6

�consists of all �-products of these basis

elements, without repetition.

Also, we can prove that

y2k

(j) =�b(k) � b�2(j+k)

��.

7 E (1; 2)� (K (Z; 3))

We have the short exact sequence of Hopf algebras

E (1; 2)� � E (1; 2)� (K (Z; 3)) f� � E (1; 2)��BP h1i

6

�v�1 � E (1; 2)�

�BP h1i

4

� � E (1; 2)� .

More explicitly,

E (1; 2) � E (1; 2)� (K (Z; 3)) f� � E (1; 2)�hhy(i) : i � 0

iiv�1 � E (1; 2)�

hhx(i) : i � 0

ii � E (1; 2)� .

Lemma 7.1 In E (1; 2)��BP h1i

6

�, we have the equation

[v1] � b(0) � b�2(j) =X

1�w�2jCw;2jb(0) � bw

for all j � 0. Also, for 0 � m � j

[v1] � b(m) � b�2(j) =X

1�w�2jCw;2jb(m) � bw.

The �est case gives us:

v�1�x(0)

�=

Xj�0

c0(0;i)y(j)

= v1y(0) + v2y(1) +Xj�2

C1;2jy(j) +Xk�1

Xj�0

c0(k;j)y2k

(j).

Lemma 7.2 In E (1; 2)��BP h1i

6

�, for i � 1,

v�1�x(i)

�= v2

i

1 y(i) + v2i2 y(i+1) +

Xj�i+2

C2i;2jy(j) +Xk�i

Xj�0

ci(k;j)y2k

(j).

To simply our notation, we de�ne

Gs(i) =X

k+j=sk�i;j�0

ci(k;j)y2k

(j).

for all i � 0 and s � i. Thus, we can re-writeXk�i

Xj�0

ci(k;j)y2k

(j) =X

k+j�ici(k;j)y

2k

(j) =Xs�i

Xk+j=sk�i;j�0

ci(k;j)y2k

(j) =Xs�i

Gs(i).

Note that the degree of Gs(i) is 2�2s+2 + 2

�for y2(s). Also, the degrees d of

a terms in Gs(i) satis�es deg y(s) < d < deg y(s+1) . For convinience, we alsode�ne G(0) = 0.

Thus, we can re-write

Corollary 7.3 In E (1; 2)��BP h1i

6

�, for i � 1,

v�1�x(i)

�= v2

i

1 y(i) +Gi(i) + v

2i2 y(i+1) +G

i+1(i)

+C2i;2i+2y(i+2) +Gi+2(i) +

Xj�i+3

C2i;2jy(j) +X

s�i+3Gs(i).

To simplify our notation, we de�ne

De�nition 7.4 Let V(k) = v2k1 v�2k�12 for all k � 1 and V(0) = 1.

1. M l(i) =

l�1Xk=0

0@l�1�kYj=1

V(l�j)

1AGi(k), for all i � 1 and l � 1;

2. N l(j) =l�1Xk=0

0@l�1�kYj=1

V(l�j)

1AC2k;2j , for all i � 1 and l � 1.

Note that if l � 1� k < 1 then we de�nel�1�kYj=1

V(l�j) = 1.

After investigating some lower degree formulas, we can re-de�ne our generatorsas follows:

De�nition 7.5 In E (1; 2)��BP h1i

6

�, we de�ne

y0(0) = y(0),

and, for all i � 1,

y0(i) = y(i) + v�2i�12 M i

(i) + v�2i�12

Xs�i+1

�N i(s)y(s) +M

i(s)

�

According to this de�nition, we can re-write

Theorem 7.6 In E (1; 2)��BP h1i

6

�, for i � 1,

v�1�x(i)

�= v2

i

1 y0(i) +G

i(i) + V(i)M

i(i) + v

2i2 y0(i+1) + V(i)N

i(i+1)y(i+1).

Furthermore, we try to generalize Tamanoi�s result to higher degree. Therefore,we change our generators a little to get a better looking formula.

De�nition 7.7 In E��BP h1i

6

�, we de�ne y00(0) = y(0) and

v2i

2 y00(i+1) = v

�1

�x(i)

�+ v2

i

1 y00(i)

for all i � 0. Also, we de�ne z00(i) = f��y00(i)

�for all i � 0.

Since y0(0) = y(0) = y0(0), by (??) and the de�nition of y

00(i), we have,

v2�y0(1) � y

00(1)

�= v1

�y0(0) � y

00(0)

�= 0.

This implies that y0(1) = y00(1). Furthermore,

Lemma 7.8 In E (1; 2)��BP h1i

6

�, for i � 1,

v2i

2

�y0(i+1) � y

00(i+1)

�= C1;2iy

2i

(0) + V(i)

0@ i�1Xk=1

0@i�1�kYj=1

V(i�j)

1AC1;2ky2k(0)1A

+V(i)

0@ iXk=1

0@i�kYj=1

V(i�j)

1AP(k)1A

wherei�kYj=1

V(i�j) = 1 if i� k < 1.

Theorem 7.9 In E (1; 2)��BP h1i

6

�, we have a set of generators y00(i) for

i � 0 satis�es

1. y00(0) = y(0),

2. v2i

2 y00(i+1) = v

�1

�x(i)

�+ v2

i

1 y00(i) for i � 0.

Moreover, in E (1; 2)� (K (Z; 3)), we have

1. E (1; 2)� (K (Z; 3)) �= E (1; 2)�hhz(0)

iias a topological power series ring,

2. v2i

2 z00(i+1) = v

2i1 z00(i) for i � 0.

References

[B82] J. Michael Boardman, The Eightfold Way to BP -operations or E�Eand All That, Canadian Math. Soc. Conf. Proc., 2(1982), Part 1, 187-226.

[RW77] D.C. Ravenel and W.S. Wilson, The Hopf Ring for Complex Cobor-dism, J. of Pure and Applied Algebra, 9(1977), 241-280.

[RW80] D.C. Ravenel and W.S. Wilson, Morava K-theory of Eilenberg-MacLane Spaces and the Conner-Floyd Conjecture, Amer. J. Math.,102(1980), 691-748.

[RWY] D.C. Ravenel, W.S. Wilson and N. Yagita, Brown-Peterson Cohomol-ogy from Morava K-theory, K-Theory, 12(2)(1998), 149-200.

[T] H. Tamanoi, Spectra of BP -linear Relations, vn-series, and BP Co-homology of Eilenberg-Mac Lane Spaces, Trans. Amer. Math. Soc.,352(2000), no.11, 5139-5178. .

[Wil73] W.S. Wilson, The -spectrum for Brown-Peterson Cohomology, PartI, Comment Math. Helv., 48(1973), 45-55.

[Wil75] W.S. Wilson, The -spectrum for Brown-Peterson Cohomology, PartII, Amer. J. Math., 97(1975), 101-123.