torsion in h-spaces - university of edinburgh

ANNALS OF 1MATHERMATICS

Vol. 74, No. 1, July, 1961 Printed in Japan

TORSION IN H-SPACES

BY WILLIAM BROWDER*

(Received June 16, 1960)

A topological space with a continuous multiplication with unit is called an H-space. The topological properties of these spaces have been investigated by many authors, in particular the homology and homotopy groups. The case of Lie groups has been investigated intensely and many interesting results have been obtained by special methods for these groups. E. Cartan [8] proved that the second homotopy group of a Lie group is zero, a result which also follows from Bott's work [4]. In this paper we obtain a new proof of Cartan's theorem, using homological methods. Unlike the pre- vious proofs which made strong use of the infinitesimal structure of Lie groups, the proof given here depends only on the homological structure and can be applied to H-spaces whose homology is finitely generated.

If X is a simply connected H-space whose homology is finitely generated, then it follows from Hopf's theorem [12] on Hopf algebras that H2(X; R) = 0 (where R = real numbers) and hence that r2(X) is finite. The argument would show that a non-zero element x e H2(X; R) has infinite height (Xn # 0 for all n) which would contradict the hypothesis of finitely generated homology. Now if r2(X) # 0, then H2(X; Z.) # 0 for some prime p. While x e H2(X; Z,) may not in general have infinite height, a slightly weaker statement is proved; i.e., that x has a property called cc-implications (see definition in ? 6) which would again contradict the hypothesis that H*(X) is finitely generated.

The above follows from a general theorem (Theorem 6.1) which gives a condition ensuring that an element will have cc-implications. Many consequences are deduced from this, particularly for H-spaces whose homology is finitely generated. If X is an arcwise connected H-space with H*(X) finitely generated, it is proved that the mod p Hurewicz homomorphism h: 7rm(X) 0 Z. ) Hm(X; Z.) is zero in even dimensions m, for all p, which implies that the first non-vanishing higher homotopy group of X occurs is an odd dimension.

The known examples of H-spaces whose homology is finitely generated seem to consist of the Lie groups, the seven sphere S7, real projective 7-space P7 and products of these. These are all manifolds. It is shown here that for an H-space X with H*(X) finitely generated, the highest dimensional non-zero group Hn(X) is isomorphic to the integers Z, and

* This paper was written while the author was a National Science Foundation Postdoctoral Fellow.

24

TORSION IN H-SPACES 25

that the Poincare duality theorem holds. This might lead one to ask whether there is an example of such an H-space which is not the singular homotopy type of a manifold.

Our method is to study certain Hopf algebras arising from an H-space X, by applying a functor called the Bockstein spectral sequence. This is a functor on chain complexes which yields a spectral sequence measuring the p-torsion of the homology. Its functorial nature gives it advantages over the classical formulation of higher order Bockstein operators. In the case of the chain complex of an H-space, the Bockstein spectral sequence is a spectral sequence of Hopf algebras, the spectral sequences for chains and cochains being dual Hopf algebras.

In ? 1 we define the Bockstein spectral sequence and list some of its properties which are proved in ? 2 and ? 3. In ? 4, we discuss the Bockstein spectral sequences for spaces, and show that for H-spaces, these spectral sequences are dual differential Hopf algebras. In ? 5 we discuss the cohomology spectral sequence for K(Zpr, n) and prove a theorem about the cohomology spectral sequence of any space. The homology spectral sequence of an H-space X is discussed in ? 6, and the results of ? 5 are used to prove Theorem 6.1, which is our main technical result. This is then applied to obtain the theorems mentioned above, including the theorem of Cartan. The Poincare duality theorem is proved in ? 7.

Note that no associativity hypothesis is required for any of these results. Some of the results of this paper were announced in [5]. I wish to thank J. C. Moore who pointed out the construction of the

Bockstein spectral sequence by means of an exact couple. This construction is also found in the work of M. Nakaoka [15]. Proposition 2.4 was suggested by J. H. C. Whitehead, who made many other useful comments. Finally, thanks are due to S. MacLane, J. Stallings, N. E. Steenrod, and R. G. Swan for very valuable discussions and suggestions.

1. The Bockstein spectral sequence

Let Z be the integers and let p be a prime.

DEFINITION. A chain complex A is a graded Z module (abelian group) (i.e., A = A A) with a boundary operator d of degree s = + 1,

d:A,-)A,, d2=O. A is called free (torsion free) if Ai is free (torsion free) over Z for each

i. Let the kernel of d = C (the cycles), the image of d = B (the bounda-

ries). The homology H(A) of A is defined by H(A) = C/B and is a graded module.

26 WILLIAM BROWDER

The Bockstein homomorphisms are defined classically as follows, where A is a torsion free chain complex:

Let x e H(A 0 Z,,), c e A a chain representing x in the sense that if j: A - A 0 Z, is reduction mod p, then j(c) is a cycle representing x. Thus d(jc) = j(dc) = 0 which implies that dc = pe for some e e A. Since d2= 0, we have d2c = p de = 0 which implies de = 0. Define the first Bockstein homomorphism 8, by 8,(x) = {je} (where {I } denotes homology class). Then 81: HJ(A 0 Z.) - H,+S(A 0 Z,) is well defined and has square zero.

In general dc = pre with e e A for some r. Define flr(x) = {je}. One can show that fir is only defined on the intersection of the kernels of E,, i < r, its value is in the quotient by the images of f,, i < r, and o = . In other words a spectral sequence E,(A) can be defined inductively by E,(A) = H(A (0 Z,,), di = 83 and Eri = H(Er) with respect to the differential Air so that fir is a differential in a spectral sequence. We now define this spectral sequence in a functorial way, using the theory of exact couples (see [13]).

An exact couple is an exact triangle

D D \ / k\ /i

G

A spectral sequence {Er} is obtained from an exact couple by defining E, = G, d1 = jk and defining the derived couple

i(D) i (D) /

k' \ jr

E2

where E2 = H(Ej) with respect to d,, i' is induced by i, k' is induced by k, and j'(i(a)) = {j(a)}. One can show these maps are well defined and the derived couple will again be an exact couple. This process leads to an inductive definition of a spectral sequence.

If we take our torsion free chain complex A and tensor it with the exact sequence 0 -+ Z -> Z Z, -*0, we get an exact sequence of chain complexes

0 - A- )A 2 A? Zv- >0

(A 0 Z being identified with A). By the usual argument the homology of


this sequence gives rise to an exact couple

H(A) H(A) M 8P~~~~O\ /i*

H(A (0 Z.,) DEFINITION. The spectral sequence associated with the exact couple (I)

is called the Bockstein spectral sequence of A mod p. One can check that this is the same spectral sequence as defined above,

with the differentials being Bockstein operators. Since A is graded and d has degree s, Er is also graded and dr has degree

s. Note that there is no filtration in this construction of the spectral sequence.

A map f: A A' of torsion free chain complexes induces a map of exact couples and hence of spectral sequences, and let the induced map be denoted by fr: Er(A) - Er(A'). The following propositions list the main properties of the Bockstein spectral sequence which we will use.

Let C and D be free chain complexes with differentials of degree + 1, and suppose Hm(C) and H1(D) are finitely generated for each m. Then Er+l(C) = Er(C) for r large enough so that one can define Eoo(C) (the spectral sequence converges).

The map j induces j( ,=j*: H(C)-)H(C?Z,) = E1. Since image j(1) c kernel dj, j(1) induces j(2). H(C) E2, etc. We define 6(r): H(C) Er by induction, since image 6(r) c kernel dr. Since Hm(C) is finitely generated for each m we can define j(o): H(C) Eo. byj(o) = 6(r) for r so large that E. = Er.

PROPOSITION 1.1. The kernel of j(oO) = T + pH(C), where T = torsion subgroup of H(C), and E(.) = j(.)(H(C)) so that E(.) = (H(C)/T) 0 Z,.

PROPOSITION 1.2. If x e Er(C), x # 0, then there is an element x' e H(C (0 Z1,r) such that x = {kex'} where k: Zpr - Z. If y generates a direct summand Zpr in H(C) then 3(r)(Y) :$ 0 in Er.

The tensor product C 0 D of chain complexes C and D is defined by (C 0 D)n = 5+'ij+=n Cf (0 D with

d(a ( b) = (da) ( b + (-1)dimaa ( (db) If a e C (0 Z, b e D?( Z, with da = db = 0, then the homology class of a (0 b in H(C (0 D 0 Z,) depends only on the homology class of a in H(C) and the homology class of b in H(D). Thus we may define a natural map f: H(C0 Z.) (OH(DO( Z.) - H(CO( D(OZ) byf(a f,) = {aXb}, where {a 0& b} is the homology class of the cycle a (0 b mod p in H(C (0 D 0 Z,,), where a e C (0 Z, and b e D ( Z1, are cycles representing a and f,

28 WILLIAM BROWDER

a e H(C (0 Zj), fi e H(D 0 Z.). The Kiinneth formula tells us that f is an isomorphism.

PROPOSITION 1.3. f induces a map f(,) of Bockstein spectral sequences and f(r): Er(C) (0 Er(D) - Er(C 0 D) is an isomorphism of chain complexes for all r > 1.

PROPOSITION 1.4. Er = Er(Hom (C, Z)) = Hom (Er(C), Zp) as chain complexes (dr = (dr)*, the adjoint of dr).

The proofs of these propositions are based on the technique of decom- posing a complex into elementary subcomplexes. In ? 2 we will develop some of the theory of chain complexes necessary for this, and in ? 3 we apply these techniques to the Bockstein spectral sequence, and prove the above propositions.

2. Chain complexes

In this section we develop some useful properties of chain complexes. NOTATION. The letters A and A' will denote chain complexes with dif-

ferentials d of degree -1 (the case of degree +1 being similar). Cn = kernel d: An -* An-l C' - kernel d: A' -* Aln-l Bn = image d: An+1 ? An -

Bn = image d: A'+, ) An PROPOSITION 2.1. If A is free, p: H(A) H(A') a map of homology

groups, then there exists a chain map f: A - A' such that f* = 9. PROOF. Since A is free, Bn and Cn are free for each n. The sequence

o - Bn, - Cn > Hn(A) - 0 is exact. Also

d (*) 0 -) Cn, > An < > Bn-1 -> 0

p

is exact and there exists a p, with dp = identity, splitting (*) since Bn, is free. Hence we have the diagram

o ) Bn- - Cn H(A) > 0

{f {f P O -~ - Bn C, - Rn(A') - 0,

where f exists because Cn is free. This defines f on Cn and by induction f is defined on Bn-1, so we may define f on all of An by using the splitting of (*). Then f commutes with d and is the desired chain map.

PROPOSITION 2.2. Given A', there exists a free complex A and a chain


map f: A A' such that f,: H(A) - H(A') is an isomorphism. PROOF. It suffices, in view of Proposition 2.1, to construct a free com-

plex having prescribed homology groups Hn. Take a free resolution of Ha,

0 - KS i Ln > Hn ) 0 . Set An = Ln + Kn-1 and d(c) = 0 if c e Ln, d(c) = in-(C) if c e Kn1.

PROPOSITION 2.3. Let f: A A' be a chain map, A free. Let An= Cn + Bn-1 be a splitting of (*). If g: Bn_ - C' is a map, define -: An -' by -(c, b) = g(b). Then (f + g) is a chain map and (f + )=f*: H(A) -

H(A'). The verification is immediate. While in Proposition 2.3 the maps of integral homology are the same,

the induced maps of homology with other coefficients may be different. Thus in constructing the maps of Proposition 2.1 we may get many different maps in homology with other coefficients for the same map of integral homology.

PROPOSITION 2.4. Let A be free, A' torsion free, and p prime. Let v

be a map of the exact homology sequence for A arising from the exact co-

efficient sequence 0 -- Z > Z : ZP - 0, into the corresponding sequence for A'. Then there exists a chain map f: A-A' such that f*=? as maps of sequences.

PROOF. By Proposition 2.1 we may choose a map h: A A' such that he* = on H(A). If we consider g -h* as a map on the short exact sequences arising from the Universal Coefficient Theorem [10, page 161] we get:

0 - Hm(A) 0 Z, > Hm(A 0 Z,,) Tor (Hmi(A), Zp) 0

vu-h* v <z-h* v-h*l a

0 > Hm(A') 0 Z, > Hm(A' 0 Z,) - Tor (Hmi(A'), Z,,) 0 0. The maps on the tensor and Tor terms are zero since q = h* on H(A). Then p - h* defines a map t: Tor (Hmi(A), Z,,) Hm(A') 0 Z, by

&(a) = a-'(p- h*)(j3-1a) where a e Tor (Hmi(A), Z,). One can check that a is well defined and a4/3 = - h*.

Consider the diagram:

Tor (.Hm-(A), Z..) nB,1 0 Z1 <4 Bm-

Hm(A') Zp AC' XZ1 CM

30 WILLIAM BROWDER

The maps are defined as follows: (i) [I comes from the sequence defining Tor

0 -- Tor (Hmi(A), Z,,) -> Bm, 0 Z,

- CM-1 (9 Z.,_ > Hm_j(A) () Z. > O;

(ii) -y' comes from the above sequence with Hm(A') instead of Hmi1(A); (iii) v and V are reduction mod p; (iv) Since p is prime and p is 1 - 1, we have that Tor (Hmi(A), Z,) is

a direct summand of Bm, 0 Z,, so there exists a map z splitting the sequence, with z-p = identity;

(v) 0' = AZ; (vi) -y'V: C- Hm(A') 0 Z, is onto and Bm, is free so that the map

#'v: Bm, Hm(A') 0 Z, lifts to a map #: Bin-i CM'. Use the splitting of (*) as in Proposition 2.3 and set f = h + A. By

Proposition 2.3 this is a chain map and f* = h* on integral homology. Then by the above remarks, on Hm(A 0 Z,), p -f* = a(a-1(9 -f*)f31)f.

Butf* = h* + A* so 9 -f* = p - h*- *, hence

a-1( - f*)i1 = a-1(9 -h - )3-1 - (a-1(9 - h*)l1) - = -a_ (a-1+1)

One can verify that alr-1-l = and thus p - f* = 0 or 9 = f* on H(A) and H(A 0g Z,).

3. Properties of the Bockstein spectral sequence

LEMMA 3.1. Let A and A' be torsion free chain complexes, A + A' the direct sum complex (A + A'), = An + A', the differential being the sum of the differentials. Then ET(A + A') = EJ(A) + E,(A').

LEMMA 3.2. Let f: A - A' be a chain map, A and A' torsion free chain complexes, and suppose f*: H(A 0 Z,) - H(A' 0 Z,) is an isomorphism. Then fT: EJ(A) > E,(A') is an isomorphism for all r.

For f* = f1 is an isomorphism on E1, hence fT is an isomorphism on E, for all r.

It follows from Lemma 3.2 and Proposition 2.2 that we may replace the chain complex A' by a free chain complex A which would be a direct sum of complexes with only one non-zero homology group in such a way that

EJ(A) - E,(A') for all r. From Lemma 3.1 it follows that this spectral sequence is the direct sum of the spectral sequences of each summand.

NOTATION. For the remainder of this section "complex" will mean a free chain complex whose homology groups are finitely generated in each


dimension. It follows that EJ(A) _-EE(A,), where A, are complexes whose homol-

logy is non-zero in only one dimension; and, in that dimension, is cyclic. In fact using Proposition 2.4 we can find maps f: A Ai, g: ea A A such that the induced maps on the exact couples are the identity map on each group. Given a map 4:: A - A' of complexes, and a similar set of maps for A', f ': A' EJ A', g': EJ A' -A' inducing identity maps on the exact couples, then define 4:': E Ai E A' by 4:' f'4g. Then the diagram

A 2 A'

,f g g,A

commutes, and hence 4:' induces the same map of spectral sequences as 4:. Thus our substitution of E Ai for A may be made in such a way as to commute with a given map.

DEFINITION. Let n and 0 be integers, 0 _ 0. The elementary chain complex A(n, 0) is defined by

A(n, 0)m O if m n, n + 1, A(n, 0)n Z with generator u, A(n, 0)n,? = Z with generator v, if 0 # 0, dv = Ou,

and A(n, O)n+? = 0

Then Hn(A(n, 0)) = Zo (where Z, = Z), H7),(A(n, 0)) = 0 for m # n. Thus we have proved:

THEOREM 3.3. One can replace a complex by a direct sum of elementary complexes which has an isomorphic Bockstein spectral sequence, and the substitution can be made in such a way that the induced maps are the same for a given map.

LEMMA 3.4. If (0, p) = (greatest common divisor of 0 and p) 1 then E1(A(n, 0)) = E.(A(n, 0)) = 0.

PROOF. E1(A(n, 0)) = H(A(n, 0) 0 Z,) = 0.

LEMMA 3.5. H(A(n, 0) (0 Zi) = E1(A(n, 0)) = E.(A(n, 0)). PROOF. H(A(n, 0) 0 Z,) is non-zero only in dimension n, so that d, = 0

for all r, the degree of d, being equal -1.

LEMMA 3.6. Let (a, p) = 1 and m ? 0. There is a map p of the Bock- stein exact couple of A(n, aptm) into the derived couple for A(n, apmn+)

32 WILLIAM BROWDER

which is an isomorphism on each term. PROOF. Set A = A(n, apm+1), A' = A(n, aptm) and denote the generators

by u and v, u' and v' in dimensions n and n + 1 respectively, with dv = apm'+1u, dv' = apmnu'. Define q: A' - A by q(u') = pu, q(v') = v. Then q is a chain map and q*(H(A')) = i(H(A)) and q* is an isomorphism onto i(H(A)).

Define s: A' 0 ZZ, -- A 0 Z, by sending u' into u and v' into v. If m > 0,. then d = 0 in A' 0 Z, and di = 0 in E1(A). Hence s induces an isomorphism s* of H(A' 0 Z,) with E2(A). If m = 0, H(A' 0 Z,) = 0 = E2(A),. so that s induces an isomorphism trivially.

Define p to be q* on H(A') and s* on H(A' 0D Zr). It remains to verify that p is a couple map.

Since q is a chain map, iq* = qi, hence i'? = pi. The class {u'} generates H(A') so it suffices to check j'p = pj on lu'}.

Now j{u'} = {u'b, (reduction mod p) so 9j{u'} = 9{u'}p {u}l,, while j'p{u'} =

j'{pu} = j'{i(u)} = {ju} = {u},. Finally &,{u'}%= 0= 8&{u}l, so it remains to show that p89{v}l

8cp{v'}p=8j{v}P. But if m >O so that {v'}lpO, then 8,{v'}p= {apmlu'}, while &,{v}P = {apmlu}. Since u{'} = {pu} the result follows.

Hence 9 is an isomorphism of exact couples and the lemma is proved.

THEOREM 3.7. Let A be a complex. There exists a complex A' and a, map 9 of the exact couple for A' into the derived couple for A which is an isomorphism of couples. If f: A - B is a map of complexes, one cant choose f': A' - B' so that 9f* =f(,-

PROOF. The first part of the theorem follows from the preceding lemma and Theorem 3.3. The existence of f ' follows from Proposition 2.4.

LEMMA 3.8. If 0 = qpf, (q, p) = 1, then

E1(A(n, 0)) = Ef(A(n, 0)), E,+l(A(n, 0)) = E4(A(n, 0)) = 0

PROOF. Apply Lemmas 3.4 and 3.6 and use induction on f. PROOF OF PROPOSITION 1.1. Apply Lemmas 3.4, 3.5, 3.8 and Theorem

3.3. - PROOF OF PROPOSITION 1.2. The first part follows from Lemmas 3.4, 3.5, and 3.8 together with Theorem 3.3.

For the second part, let a: Zpr -- Hn(C) and /8: Hn(C) - Zpr be maps which make Zpr (generated by x) a direct summand, so fla = 1. By Prop- position 2.1, there are chain maps f: A(n, pr) C, g: C - A(n, pr) such thatf* = a, g* = /, so that ga*f = 1. It follows that E,(C) has Er(A(n, pr)) as direct summand and the statement j(r){u} # 0 in E,(A(n, pr)) follows from Lemma 3.8. But f*{u} = y, hence j(r)Y # 0 in Er(C).


PROOF OF PROPOSITION 1.3. The Ktinneth formula over Z, says that E1(C 0& D) = H(C (0 D (0 Zp) = H(C 0& Zv) (0 H(D (0 Zv) = El(C) ? E1(D) and it is easy to verify by direct computation that d1 is a derivation. Then the result follows by induction using Theorem 3.7 and Lemma 3.9 below.

If A is any complex, let A' denote a complex according to Theorem 3.7 which has its exact couple isomorphic to the derived couple for A. Set C= C1, D= C2 for convenience.

LEMMA 3.9. C' (0 C' can be chosen for (C1 C2)'. PROOF. It will suffice to prove it for C,= A(n, mpf), C2 A(s, tpg) where

(sm, p) (t, p) = 1. Then C =A(n, mpf-1), C =A(s, tpg-l). We may assume f > 1, g > 1, for otherwise E2(C1 0 C2) = 0 = E1(C' 0 Cf).

We denote the generators of C, by ul, vi, with dv1 = mpfu1, dv2 = tpu2, corresponding generators of C' denoted by u', v', etc., and let us assume f _ g. Let

7 = least common multiple of m and t = am =,at 8 = (mpf-t, t) = (mpf-g) + rYt, x = a(V1 0 U2) -_a 0-g(U 3 V2)

Y = 5(V1 0 U2) + (-1)dimuy(U1 0 V2) .

Then {x, y} is a basis for (C1 0 C2)8+,+1 and similarly putting primes on x, y, ui, vi in the above equations we get elements x', y' which together form a basis for (C' 0 C')8+,+?. Further dx = dx' = 0, dy = 3pg(ul 0 U2),

dy' = 8pg-l(U8 0 us). Define q: CO? Cf C1 C2 by

q(,uf 0 U2) = AU1 (8 U2) I

q(x') = px,

q(y') = Y q(vf 0 v2) = V1 0 V2-

Then q is a chain map and q*: H(Cf 0Q C -) iH(Cl 0 C2). Define chain maps si: Cf 0 Zp C, 0 Z,, by si(u4) = ui, si(v) vi. Then

if s = S1 0 S21

s*: H(C 0 C2 0 Zp) -H(C1O C2 O ZP) E2(Cl 0 C2).

Define p = q* on H(C t0 Cf), 9 = s* on H(C0 C (0 ZP). One can verify, as in the proof of Lemma 3.6, that p is a map of couples,

hence an isomorphism of couples which proves the lemma. PROOF OF PROPOSITION 1.4. Hom (A(n, 0), Z) is an elementary complex

with homology group Z0 in dimension n + 1 if 0 ? 0 (the differential now being of degree + 1). It is easy to verify the proposition directly in this case, the general case following from Theorem 3.3.

34 WILLIAM BROWDER

The following theorem shows one relation between E, and p-torsion in H(C) which we use later.

THEOREM 3.10. H(C) has no p-torsion if and only if E1(C) = E.(C) in the Bockstein spectral sequence mod p.

PROOF. As usual apply Lemmas 3.4, 3.5, 3.8 and Theorem 3.3. In ? 6 the following theorem will prove useful.

THEOREM 3.11. If x e E,(C), then d x = 0 for all j ? r if and only if x = i(r)X'for some x' e H(C).

PROOF. The proof is obvious for C an elementary complex. The theorem then follows by applying Theorem 3.3.

4. Bockstein spectral sequences for spaces; H-spaces; Hopf algebras

Let X be a space, C*(X) the singular chain complex of X. We will make the assumption that any space X under discussion has its singular homology groups HJ(X) finitely generated for each n.

DEFINITION. The Bockstein spectral sequence of C*(X) mod p is called the Bockstein spectral sequence of X in homology mod p and is denoted by E(r)(X) (or where there is no ambiguity, simply E(r)). If f: X-) Y is a continuous map, the induced map of spectral sequences is denoted by f(r): E(r)(X) E(r)(Y). Similarly if C*(X) = Hom (C*(X), Z) is the cochain complex of X, we have Er(C*(X)):= E(r)(X) (or simply E(r)) is the Bockstein spectral sequence of X in cohomology mod p, and induced maps are denoted by f(,). These two spectral sequences are dual by Proposition 1.4.

We will often omit mention of the prime p.

DEFINITIONS. Let K be a field A, A, B, and C modules over K. (1) An algebra A is a graded module and a map p: A 0 A > A. (2) A co-algebra C is a graded module and a map A: C - C 0 C. (3) A unit for an algebra A is a map (: K - A (K is graded by K = K0)

such that '() 0 1) = (1 0 () = identity where K 0 A and A 0 K are identified with A.

( 4 ) A co-unit for a co-algebra C is a map C - K such that (s 0 1)> =

(1 0 s)e = identity, where K 0 C and C 0 K are identified with C. ( 5 ) An augmentation of an algebra with unit is a map s: A K of

algebras with unit. ( 6 ) A co-augmentation of a co-algebra with co-unit is a map (: K > C

of co-algebras with co-unit (K is made into a co-algebra by sending 1 1 0 1).

Let T: A (0 B B (B A be given by T(a 0 b) = (_1)dimabdi bb (0 a.


(7 ) If A and A' are algebras, then A 0 A' is the algebra defined by A: (A 0 A') 0 (A 0 A') A A0A', 9 = (p q ')(1 ? T ? 1).

(8) A Hopf algebra is a module A together with maps q: A 0 A A, A: A A0 A, (: K- *A, s: A K such that

(i) A is an algebra under p with unit (2, augmentation s; (ii) A is a co-algebra under J with co-unit e, co-augmentation (; (iii) J is a map of augmented algebras with unit.

(9 ) A differential algebra (co-algebra) is one equipped with a differential d with d2 = 0 such that dq = qd(fd = d*) and ed = 0, dy = 0.

(10) A differential Hopf algebra is a Hopf algebra with a differential, which is both a differential algebra and a differential co-algebra.

(11) An algebra (co-algebra) is called associative (co-associative) if -P(1 0 A) = P(P 0 1) ((* 0 1)* = (1 0 00).

(12) An algebra (co-algebra) is called commutative (co-commutative) if pT = p (T*= *).

(13) An element x e C, a co-augmented co-algebra, is primitive if Ax x?1 + 1?x. (Note for any ye C, qry=y11+l1y+ E yjy', where yi, yIe kernel e.)

(14) An element x e A, an augmented algebra, is decomposable if x e -p(A 0 A), where A = kernel of s.

Algebras, co-algeras and Hopf algebras are studied in [14]. We state some properties without proof. The proofs are routine verifications. All K modules will be assumed finitely generated over K in each dimension, and for any K module M we let M* = Hom (M, K).

LEMMA 4.1. A is an algebra if and only if A* is a co-algebra. The dual of any property P is property co-P, such as associative dual to co- associative etc.

LEMMA 4.2. A is a Hopf algebra if and only if A* is a Hopf algebra.

LEMMA 4.3. If A is an algebra, x e A is not decomposable, then there exists Y e A*, such that x is primitive and x-(x) + 0. Further, any primitive ye A* annihilates all decomposable elements.

LEMMA 4.4. Let A be a differential algebra. If x, y e A, dx = y, then z decomposable implies y decomposable. Let C be a differential co-algebra. If x, y e C, dx = y, then x primitive implies y primitive.

LEMMA 4.5. Let A be a differential algebra (co-algebra). Then H(A) is again an algebra (co-algebra) and all algebraic (co-algebraic) properties mentioned above are inherited by H(A), such as associativity (co-associativity).

For a space X, let A: X-) X x X be the diagonal map A(x) = (x, x),

36 WILLIAM BROWDER

i: x0 -, X the injection of the base point, c: X x0 the collapsing map.

PROPOSITION 4.6. (1) A(r): E(r) (X) ( E(r) (X) E(r) (X) makes E(,) into a differential alge-

bra with unit (= C(r), augmentation s = i(r) It is associative and commutative.

(2) A(r), E(r)(X) E(r)(X) 0 E(r)(X) makes E(r) into a differential co-algebra, etc.

(3) These are dual objects, i.e., E(*) = E(r). PROOF. The Eilenberg-Zilber theorem [11] together with Proposition

1.3 shows that A induces maps as indicated, and the properties follow easily. Proposition 1.4 implies (3).

DEFINITION. An H-space is a space X with a map e: X x X X and an element e e X such that 7(x, e) = y(e, x) = x for all x e X.

One might require that 7( , e) and 71(e, ) as maps of X - X be simply- homotopic to the identity map, instead of equal to it; or, in other words,, that e is a "homotopy unit", rather than a unit. If Xis a polyhedron, then it follows easily from the homotopy extension theorem that if / has a homotopy unit e, then 7 is homotopic to 7', with e a unit for A/'. If X is not a polyhedron, since we are only interested here in the singular properties of X, we could look at the singular polytope of X, which is an H-space of the same singular homotopy type as X.

A useful definition that we might employ is an adaptation of Hopf's; original definition [12]:

DEFINITION. A space X is a called a homology H-space modulo p, where p is prime, if there is a map -: X x X-* X such that for an element e e X, the maps 71(x) = (e, x), 72(X) = 7(x, e) have the property that (,)*. and (72)* are automorphisms of H*(X; Zr). Xis called a homology H-space if (7Y)* and (72)* are automorphisms for all coefficients.

Though we talk of H-spaces in this paper, our results would be valid for homology H-spaces with minor modifications in our arguments, and the theorems that use only one prime p (such as Theorems 6.1, 6.8, 6.9, 6.10) would be valid for homology H-spaces modulo p.

PROPOSITION 4.7. If X is an H-space then E(r)(X) and E'r)(X) are dual differential Hopf algebras, where y induces the co-algebra structure of E(r) and the algebra structure of E(r).

PROOF. One proves, as in Proposition 4.6, that 7 induces algebra structure in E~r) and co-algebra structure in E(r) . Condition (iii) of the definition of Hopf algebra (Definition 8) above, follows from the fact that A is a map of H-spaces, where if (a, b), (c, d) e X x X the product is defined by


(a, b)-(c, d) = (ac, bd). We will give a simple application of Proposition 4.7 based on the follow-

ing algebraic theorem of Milnor and Moore (see [14, Proposition 4.23]).

THEOREM 4.8. (Milnor and Moore). Let A be an associative, commutative Hopf algebra over Z,, P(A) = the primitive elements of A, D(A) = the decomposable elements of A, Q(A) = A/D(A). Let a: A A be defined by ~(x) = x". Then the sequence:

0 > P(tA) >P(A) Q(A) -is exact. In other words if ye P(A) n D(A) then y = t(u) u for some u e A.

We use Theorem 4.8 to prove a converse to a theorem of Borel (cf. [1, p. 143, remark 2]).

THEOREM 4.9. Let X be an H-space, Hi(X) finitely generated for each i, with H * (X; Zp) = lA(xi, * x * Xn, . * *), the exterior algebra on the odd dimensional generators xi. Then H*(X) has no p-torsion.

Theorem 4.9 follows from Proposition 4.10 below and Theorem 3.10.

PROPOSITION 4.10. Let A be a differential Hopf algebra over a field K -with differential d of degree + 1. If, as an algebra, A - A(x,, * * *, x - *), the exterior algebra on odd dimensional generators xl, Xn., xn, ..., then Id 0- and H(A) = A.

The proof is based on the following lemma.

LEMMA 4.11. If Ais a differential co-algebra over afield Kwith A0=K, d(Ai) = 0 for i < n, then d(An) c P(A).

PROOF. If A: A - A 0g A is the co-multiplication in A, x e An, then Ax = x 0 1 + 1 0 x + E xi 0 xi, where xi, x' are of dimension < n for -all i. Hence dxi dx' 0 and *(dx) = d(*(x)) = dx 0 1 + 1 0 dx so -that dx e P(A).

PROOF OF PROPOSITION 4.10. Assume by induction that d(A) = Ofor i <n. Note that d(A) = 0 since d(l) = d(1.1) = 2d(1) hence d(l) = 0 and d(Aj) = 0 since A,= K. Let 1= d(An). They by lemma 4.11, I c P(A). By Theorem 4.8, since ptl powers are zero in an exterior algebra, it follows that no element of I is decomposable. Hence if I # 0, I is in an odd dimension, ,since any even dimensional element of A is decomposable. Since degree Wof d = + 1 we have that n is even and An c D(A), so by Lemma 4.4, I c D (A) and thus I = 0 and d = 0. q.e.d.

It will be convenient to have the following characterization of E(o).

THEOREM 4.12. Let X be an arcwise connected H-space with Hi(X) finitely generated for each i. Then E(oo) - (H*(X)/Torsion) 0 Z. as a

38 WILLIAM BROWDER

Hopf algebra. PROOF. By Proposition 1.1, E(.- (H*(X)/Torsion)(0Z as a Z,-module-

Then Theorem 4.12 follows from the fact that j(., is a homomorphism of' algebras and commutes with the diagonal map.

COROLLARY 4.13. Let X be as in Theorem 4.2, and suppose H*(X; Zj is a finite Z, module. Then E(oo) = A(x, *. * *, xn), dim xi odd.

PROOF. A theorem of Borel [2, p. 405] says that H*(X)/Torsion A(xl, *.., Xn).

COROLLARY 4.14. Let X be as in Corollary 4.13. Then there are no even dimensional primitive elements in E(-).

PROOF. All even dimensional elements of E(,,) are decomposable. Hence by Lemma 4.3, any primitive y e Em' annihilates all even dimensional. elements and so is not even dimensional, since E(-) = E(*oo

Corollary 4.14 will be used later in ? 6.

5. E(r) (K(Zpry n)) The principal result of this section is Theorem 5.4 which is needed for

the applications in ? 6. This theorem, describing a special property of the differential in the cohomology Bockstein spectral sequence, is proved by the method of the universal example, which is in this case the Eilenberg- MacLane complex K(Zr, n). In the course of the investigation we deduce the algebraic structure of this example (Theorem 5.5).

Let A(M) = exterior algebra on the set M, P(M) = polynomial algebra on the set M, over the field Zp. If M is graded P(M) and A(M) are graded by the rule that, if dim x= m,

and dim y n n, then dim xy = m + n. Define the following differential algebras over Zp:

Al = A(x) ? P(y) dx = y, A2= A(x) 0 P(y) dy = x.

LEMMA 5.1. A1 is acyclic, (Hm(Ai)= O, m#O) and H(A2)= A(W) 0 P(Z) where W= {xyP-1}, Z {yP}.

K(r,n) will denote a space such that Wn = , ,i= O. i zn. E(r)(K(Zprn )) will always denote the mod p cohomology Bockstein spectral sequence.

Let c=identity e Hom (Zpr, Zpr) = H(K(Zpr, n); Zpr) , c= j* , j: Zpr-,Zp. Then c e Hn(K(Zpr, n); Zp) = E(1,) is a cycle under d (,) for k < r and d (r)a = '2 where by abuse of language we denote elements in E(1) and E(r), by the. same letters, (En) - E(9l, E = E)yE so we also takerC e Hn+(K(Zpr, n); Zp)).

We denote by 8P the Bockstein co-boundary associated with the coefficient sequence 0 ) Zp - p2 -ZP ) 0.


Let p be an odd prime. H. Cartan [9] has computed H*(K(Zpr, n); Zp) and found that it is the free commutative algebra over Zp, with generators given by so called "admissible sequences" of Steenrod operations on c and 7. If a sequence S is admissible, either S = 8pS' with S' admissible or ,/pS is admissible. Hence since 3p = d(,) in E(1,), all the generators are, paired off under d 1,) except for c and C when r > 1. Hence we may write

H*(K(Zpr, 2n), ZP) = P(c) 0 A(7)) 0 0,AG,

H*(K(Zpr, 2n + 1); Zp) = A(c) 0 P(r2) 0 0g AG,

where G runs over generators of H* (except for c and () and

AG= A(G) ( P(i3pG) if dim G odd, = P(G) (0 A(i3pG) if dim G even.

Thus each AG is closed under d(,) = 8p, as is A(c) (0 P(7)) or P(c) 0& A(7) also. Hence we have

E(2)(K(Zpr, 2n)) = H(P(c) 0 A(C)) 0 @GH(AG)

E(2)(K(Zpr, 2n + 1)) = H(A(c) (0 P(r7)) (0 0G H(AG)

by the Kiinneth formula over a field. Let ?Fd: HnHn+2i -P` be the itlSteenrod operation. Consider K(Z r, 2n).

The lowest dimensional generator G is G = p'e, dim ple = 2n + 2(p - 1) > 2n +4. Hence Hm(A,) = 0 if m< (2n +4)p by Lemma 5.1, so (0G H(AG)),= Q if m < (2n + 4)p. Let m < (2n + 4)p. Then E Hm(P(c) 0 A(C)). But d(jc = d(j)r = O if j < r and d(rTc =(. Hence En, =E En, and E~Mr1 =P({eP}) 0& A({cP-?}) by Lemma 5.1. H. Cartan has shown that H2np(K(Zpr, 2n); Z) has a summand Zpr+l, but no factor Z or Zp, for k > r + 1. It follows that E(2n 2- 0, so d(r+l){CP} # 0 and hence we have:

PROPOSITION 5.2. Let c e E2(K(Zpr, 2n)), C = dl(rat p # 2. Then

d(r+1,CP- 0 , cc Zp

The case of K(Z2r, 2n) is somewhat more complicated, owing to the fact that squares of odd dimensional elements are not zero, and squares of odd dimensional generators are Bocksteins of other generators. If dim x = 2m + 1, then x2= Sq2m+lx= Sql Sq2m x and Sq'= ,832. If x = Sq'c, I= (al, a2, ... ) admissible (see [16]) then the sequence I' = (2m, al, a2, *...) is admissible, so SJ'c is a generator. Thus some generators are not paired off with generators but with squares of odd dimensional generators. But if dim x = 2m + 1, H(A(Sq2mx) (0 P(x2)) = 0 so that in E(2) these phenomena disappear; odd dimensional elements have square zero. Thus if r > 1, we may make an identical argument as in the case of p odd, for we get the same types of differential algebras.

40 WILLIAM BROWt)ER

If r 1, then we have Sqle = C and Sql(c) = 72 # 0. But Sq'Sq27 = Sq~n+,72 = 2 also, so that Sql(Sq2n7 + cry) = 0. Also Sql(c2) = 0 and c2 is the only possible non-bounding cycle in En(4) since other elements in this dimension are generators, which are either zero in En(2) or are not cycles under d(l). Similarly (Sq 2n + cry) is the only possible non-bounding cycle in E4,+l and since H4n(K(Z2, 2n); Z) contains a summand Z4 but not Z21, j > 2,

(4n= 0, and we have:

PROPOSITION 5.3. Let ce E 2n (K(Z2r, 2n)), d(rTe =). If r > 1, d(T+l){c2} = {cry} / 0 while if r = 1, d(2{c2} {= Sq2n) + cry}.

Let E(,) be the cohomology Bockstein spectral sequence mod p of a space X.

THEOREM 5.4. Let x e E 2, y y = d (rx =t: 0 and if p / 2 or r # 1, suppose {xP-1y} # 0 in E(+l,. If p = 2 and r = 1, suppose {Sq2ny + Xy} # 0 in E(2). Then xP / 0 and d(c+,){xP} = c{xP-1y} 0 if p # 2 or r + 1, and d(2){x2}= {Sq2ny + xy} if p = 2, r = 1.

PROOF.1 By Proposition 1.2, there is an element x e H2n(X; Zpr) such that {k,,-} = x e E(,) where k: Zpr-* Zp. Then there is a map f: X K(Zpr, 2n) such that f *(c) = x, where Y = identity e Hom (Zpr, Zpr) = H2n(K(Zpr, 2n); Zpr). (This is true if X is a polyhedron. If not, work with the singular polytope of X, which is of the same singular homotopy type as X.) Then c = kJe and f *(c) = f *(k*e) = kef *(c) = k. Then

f(r) ( X) = x , f(r) (C) = f(r) (d(r)e)= d(rCf(r)(c) = d(rCX = y

Since f(r) is a ring homomorphism, we have f(r (splt) - x= y (and fxP (Sq2n) = Sq2ny, since f(1, = f *) Since by hypothesis {jxP1y} # 0 in E(r+l), we have f(r+1)jeP-1) - {x=P-y} 1 0, so thatf(r+ld(r+l){cP} j 0 impliesf(r+)l{cP} = {XP} # 0 and

d(r+l)IXPI = d(r+lf(r+l1{cP} = C{XPy}.

If p 2, r = 1, d(2){Sq2n7 + c)2} = {Sq2ny + Xy} / 0 implies f(2){c2} = {x2} # 0 and d(2){x2} = {Sq2ny + xy}.

Theorem 5.4 gives us a hold on the differential d(r) in E(rT which we will exploit in the next section. We close this section with a description of E(r)(K(Zpr, n)) based on Theorem 5.4.

Let A be a graded algebra over Zp. Define sr: A A, r non-negative integer as follows:

1 J. F. Adams has pointed out to me that this theorem can also be proved by a direct cochain argument. This proof shows that c = 1. Such a proof of essentially the same theorem occurs in a paper of T. Yamanoshita [18]. This approach will be exploited in [6].


t0(x) = x sr(x) = 0 if r > 0, dim x odd, sr(x) = (r-1(X))P if r > 0, dim x even.

If M is a ZP module, let M+ = E, M2i, M- = , M2i+1. Define 91(M) = A(M-) 0 P(M+). Then from Lemma 5.1, Theorem 5.4 we have:

THEOREM 5.5. E(r+l)(K(Zps, n)) = _F({1rG}, {dtrG}, c, ry) if 1 < r < s; E??l(K(ZPS, n)) = J({trG}, {dTG}, t dr+l-s, d1r+l-sc) if r > s, where c, rY, {G}, {131G}, are a set of generators for H*(K(Zps, n); Zp) and d =d(r+l) is given by Theorem 5.4.

This result also follows directly from the results of H. Cartan.

6. Homology of H-spaces; consequences

In this section we prove the main theorem of the paper and deduce some consequences, and in particular the theorem of E. Cartan.

We will never make special assumptions of associativity. In case of non- associativity, define by induction Xn = Xn-i. x, x= 1.

If Mis a Z, module, M* = Hom (M, Zp), and if a e M*, b e M, we will denote by a(b) the value of the homomorphism a on b.

Let A be a Hopf algebra over Z,, A* its dual Hopf algebra.

DEFINITION. An element x e Am is said to be of oo-implications if there exists a sequence x,, x1, ..., x, ... , with xo = x, xi e Ampi, xi # 0, such that for each i either

(1) xi1,= xP, or (2 ) there exists xi e A* such that (xj) # 0 and xP(xi+1) # 0. This is a slightly different definition from that given in [5]. In case A

and A* are both associative, they are equivalent. The existence of an element of o-implications implies A is infinite dimensional.

We are now in a position to state the main result.

THEOREM 6.1. Let X be an arcwise connected H-space, with H%(X) finitely generated for each i, and let {Earn} be the Bockstein spectral sequence of X in homology mod p. If x e E2(r) is a primitive element, and x = d(ry / 0, then x has oo-implications.

(Note that this is stronger than Theorem 1 of [5] in that we no longer require y to be primitive.)

We will need the following six lemmas. LEMMA 6.1. Let A be a Hopf algebra, #: A - A 0 A the diagonal (co-

algebra) map, x a primitive element in A2m. Then

+(xn) = n xi .

42 WILLIAM BROWDER

PROOF. The usual proof of the binominal theorem with attention to& parentheses.

LEMMA 6.2. Let A be a Hopf algebra over Zp, *r: A - A (0 A the diagonal map, A* the dual Hopf algebra, so that l*r*: A* 0 A* A* is its multiplication. Let x be a primitive element in A2m, x- e A*. Then

x-n(Xn) = n ! (x-(x)) n

PROOF.

-n(Xn) = (#*(xn-, (0 X))(Xn) = (Sn 1 (0 x)(#Xn)

(x ? X) (n xn1 0 x) (xnl1 0 X)(nxn1 0 x)

= n(X n-l(xn-1). X(x)) = n!(X(X))n

by induction.

LEMMA 6.3. Let A be a differential Hopf algebra over Z,, d of degree -1, A* its dual differential Hopf algebra, A A 0 A -)--> A, and AO = ZP. Let x # 0 be a primitive element of A2n, x = dy, y e A2n+1, and let x- e A2n such that x-(x) / 0 and set d*x- = -. Then (x-P-ly-)(xP-1y) / 0.

PROOF. First note that y(y) = (d*x)(y) = x(dy) = x-(x) 0. Then (XP-1j)(xP-1y) = (fr*(-P-1 0 y-))(xP-1y) and

( 1 ) (PXP-1 0 Y-)(#(XP-ly)) = (xP-1 0

Y-)(#(xP-l)#(y))

recalling that by definition J is a map of algebras. Now by Lemma 6.1,. L= jP ( 1)xP-l-i(xi; and let #y = y!(&yi + y(3l + 1l(&.

Since y- annihilates any element not in dimension 2n + 1, expression (1), reduces to

(2) (xP-1 0 )(XP 0 y + (p - 1) 2k xy 0 xy)

where k is such that dim Yk 1. Then

(XP-1 0 y-)(xP-1 0 ) -= x -(x -1).Yi(y) / 0

by Lemma 6.2 and above remarks. We show that the value of P-' 0 ( on the second term in (2) is zero by showing y-(xz) 0 if z e A1. For, yj(xz) = (d*x-)(xz) = x-(d(xz)). But x = dy, so that dx d(dy) = 0, so that- d(xz) = xd(z). If dz / 0 in AO, then there is an element w e AO such that. d*w / O in A1. But w = oa 1 where a e Z, since AO = Zp, and d(1) = d(1 1) d(1) * 1 + 1 . d(1) = 2d(1), so that d(1) = 0. Hence d(w) = d(ac 1) = ad(1) 0. Hence dz = 0, d(xz) = 0 and ?(xz) = x-(d(xz)) = x(O) = 0, and the lemma is proved.

LEMMA 6.4. Let A be a differential Hopf algebra, x e H(A), which is


a Hopf algebra. If {y} = x, y e A and xP / 0 then yP / 0. Moreover if x has oo-implications in H(A), y has oo-implications in A.

PROOF. By definition xP {yP}, hence yP = 0. Using this argument at each step, we get the second statement, noting that if x-(x) # 0 and {yj} =x-, then y(y) # 0 (x- e H(A*) (H(A))*, y- e A*).

LEMMA 6.5. Let A be a differential Hopf algebra over Z,, x a primitive element in A2~n XP = O dy = x. If {xP-1y} # 0 in H(A), then it is primitive.

PROOF. Let #: A - A 0 A be the co-algebra map of A, A: H(A) H(A) 0 H(A) the co-algebra map of H(A). Then I{xP-1y} = j(xP-1y)} Since x is primitive d(#y) = *J(dy) = J(x) = x 0 1 + 1 0 x. Hence

d(*(y))-Y (y 1-1 ( y) = .O* By Lemma 6. 1, *(XP-1)= (P7 1 )xP-i-i (9 xi

and (P , 1) (-1)1 modp. Then

*r(X-1y) = (#xP-) (#Y)

=(E19-1 (-l1)ixp-l-i ()X)Y()1+1()y+(rg 8 8 ) (irlO)oxP i0x)(y0 1 + 1 0Y + (M() - yOl - l0y)) x-y 0 1 + 1 0 xp-ly + Li-1 (_ () 0 xiy

+ i-1 (- 1)ixP--iy (0 xi + (#xP-1)(#y - Y 0 1 - 1 08)

=xpy 0 1 + 1 0 xP-1y + d(EP=-j 0)'+,( P i x1-1 y))

+ d((fq(xP-2y))(*y - y 0 1 - 10y)) .

Hence {#(xP-1y)} = {xP-'1y} 1 + 1 0 {xP-1y} and {xP-1y} is primitive. Besides the purely algebraic lemmas above, we will also need the follow-

ing:

LEMMA 6.6. Let X be an H-space, p: X x X X. Let x be a primitive element in H2n(X; Z2), Y e H2n+l(X; Z2) and z e H2n+l(X; Z2). Then

(Sq2 Z)(ti) = 0.

PROOF. (Sq2 z)(x)= (9* Sq2Xz)( 0 i) = (Sq2n *z)(x 0 H) Let p*z zi 0 z' with dim zi + dim z' = 2n + 1. By the Cartan formula we have

Sq 2(Zi 0 Z) - = +2f Sq zi 0 Sqqzi .

Now if a > dim zi (or 1 > dim z') we have Sq"zi = 0 (Sqfz' = 0). Hence

Sq2n(zi 0z) = Sqczi 0 Sql-1z! + SqC'zi 0 Sqdz, where c = dim zi, d = dim z, c + d 2n + 1. But Sqczi (zi)2 and (zi)2(X) = 0, since x is primitive (Lemma 4.3). Hence

(Sqczi 0 Sq lz9)(x 0 ) = 0.

Also Sqdz' = (z!)2 which is even dimensional so (Sqdz!)(-) = 0, so that

(Sqclzi 0) Sqdz!)(x 0 i) 0. Hence (Sq2n(zf 0 z'))(x 0 ?) = 0 for all i so

44 WILLIAM BROWDER

that (Sq2nz)(x-) 0. PROOF OF THEOREM 6.1. To prove the element x e E r' has oo -implica-

tions we must construct a sequence of elements x = x0, xl, *. , in E(r). We shall prove the theorem by showing we can always construct the next step in the sequence. However, it will often turn out that the properties of xi,1 in E~r) are not good enough to apply the same argument as before; for instance, xi,, may be neither primitive, nor a boundary in E~r'. Our method of construction will nevertheless insure that the image of xi,, will have these properties in E r-g) for some s, and we will continue constructing elements in E r+. Then by virtue of Lemma 6.4, representatives of these elements will be the terms of the o-implication sequence in ETr'.

We may assume that xO = 0. For if not, x" is again primitive (see Lemma 6.1) and x" = d(r)(xPl-y). Hence we may start with x" instead of x, and if we never reach a zero pth power, then we have constructed our oo-implication sequence.

We will find an element x1 e E2'nr) such that x-(x1) / 0 for any x e E 2n

such that x(x) # 0. The element x1 will neither be primitive nor a boundary in E (r'. But {x1} will have these properties in E (r+l), so that we may proceed with the argument in E'r+)1 in the same way.

First we will show that {JP} / 0 in E(r+,) by showing {jP--l(d(rZ)} # 0 in E(r+,) and using Theorem 5.4 (with the appropriate modification in the argument if p = 2 and r = 1.)

If xe E 2n is such that x(x) #0, then set y-= dTr'x. Then i-(y) = (d(r~x)(Y) = X(d'r)y) =x-(x) #0. Lemma6.3 implies that x &-1y(xP-1y) #0. Nowd(r)(x --y-) = (p-1)x -2'yi220ifp 2orr>1. Forifp#2then-2 = 0because y is odd dimensional, (being in the commutative cohomology spectral sequence algebra). If p = 2 and r > 1 then -2= jSq2n+1 z = jSq1 Sq 2nZ = Oin E(2, where z e E(1,) is such that {z}j in E(rT . If p=2, r = 1, then d(1,)(Sq2ny +--) = y2 + Y2 = 0 and (Sq2 y + x-y)(xy) / 0 by Lemmas 6.3 and 6.6. Now if x y = d(r)z then

(xp-'Y)(vo-ly) = (d(r)z)(x"1y) = z(d(r)(xP-ly)) = Z(Xp) = Z(0) = 0

which is a contradiction. Similarly (Sq2n- + y Y) # d(r)z. Hence {xP-1Y} / 0 in E(r+l), or if p = 2, r = 1, {Sq 2n+ + --I 0 in E(2) . Therefore, by Theorem 5.4, UP # 0 and d(r+l{xP} =c{xPy}, (c # 0 in Z=), or ={Sq28 + y} if p = 2 and r = 1.

Next we will show that there is a primitive element a e E2 (r +1) with {XP}(a) # 0 and a = d(r+~1b. Then starting from a we will continue building a sequence in E(r+l) the same way. If x1 e E(r) such that {x1} = a e E(r+-1 then X"(xj) # 0 so that xl can be taken as the next step in the oo -implication sequence for x and by Lemma 6.4, the existence of the next step


in the sequence in E r+?1 implies its existence in E r,. Hence this construction never terminates and therefore yields an o-implication sequence.

First note that d (r(XP-1y) - xP = 0. If xp-1y = d(r)z then (X1P )(xP-1y) = (X-1- )(drz) (d(rj (r-1-))(z) = 0 since xP-ly is a cycle, or if p = 2, r = 1, (Sq2n- + x-)(xy) = 0, similarly. But this is a contradiction so that xP-1y + d(r)z and b = {xP-1y} / 0 in E'r+T1. Hence {fP-1`}(b) # 0 and therefore {fx`}(dr+1b) = c{fxP-X-}(b) # 0, so set a = dTr+'b. Then if a is primitive in E(ri1l we are done; whereas if b is primitive, then a = dTr+lb is also. But b primitive follows from Lemma 6.5, which completes the proof.

REMARK 6.7. The co-implication sequence constructed above has a slightly stronger property than necessary; i.e., if xi = 0 then P(xj+?) 0? for any x-j e E(r) such that x-i(xi) # 0. In particular if x = d(r)y, x primitive in E,(nr) xP = 0, then x-P # 0 for any x- e E(r) such that x-(x) A 0.

THEOREM 6.8. Let X be an arcwise connected H-space with Hi(x) finitely generated for each i, and suppose that there exists an N such that Hi(X; Zp) = 0 for i > N. If j.: H.(X; Z) - H.(X; ZD) is induced by j: Z > Zp (reduction mod p) then the image of j,, contains no even dimensional primitive elements.

PROOF. If a primitive element x e image j*, then it is a cycle under d~T' for all r. By Corollary 4.14, {x} = 0 in Em', and thus {x} = d(r)y in E') for some r. Then by Theorem 6.1, {x} is of co-implications, hence Hnpi(X; Zp) # 0 for all i, which is a contradiction.

THEOREM 6.9. Let X be as in Theorem 6.8. If h: wm(X)O&Zp >Hm(X; Zp) is the mod p Hurewicz homomorphism, then h = 0 for even m.

PROOF. All cycles on the sphere are primitive and integral. Hence image h consists of primitive cycles and image h c image j*, so by Theorem 6.8, (image h)2, 0.

THEOREM 6.10. Let X be as in Theorem 6.8. Then the first non-vanishing group Wrm(X) 0 Zp for m > 1, occurs for m odd.

PROOF. If X is simply connected, then the Serre-Hurewicz theorem [17, Ch. V, ? 2, Proposition 2] states that h is an isomorphism for the first non-vanishing group. Hence the result follows from Theorem 6.9.

If X is not simply connected, then its universal covering space X is an H-space, and w1(X) act trivially on the homology of X, (see [17, Ch. IV Corollary to Proposition 3])1. It follows that Hi(X) is finitely generated for each i (see [17, Ch. III, Proposition 1]). Then Hi(X) = 0 for i > N follows

In case X does not have sufficient local connectivity to have a universal covering space, we may replace X by its singular polytope which is a polyhedron with the same homology and homotopy as X, and is also an H-space.

46 WILLIAM BROWDER

from the Corollary of [71 and we may apply the above argument to X. The result now follows from the fact that Wm(X) 0 Zp W im(X) 0 Z, for m> 1.

THEOREM 6.11. Let X be an arcwise connected H-space with H,(X) finitely generated for each i, and H,(X) # 0 for only finitely many i. Then the first non-vanishing higher homotopy group Wm(X), m> 1, occurs for m odd. In particular, wr2(X) = 0.

PROOF. Apply Theorem 6.10 for all p. Theorem 6.11 is a generalization of E. Cartan's theorem that w2(G) = 0

if G is a Lie group, and yields a new proof of his result. As a consequence of Theorem 6.1, we obtain a theorem about the co-

homology Bockstein spectral sequence. THEOREM 6.12. Let X be as in Theorem 6.1. If x e E,, x primitive

and d(rTX = y * 0, then x has oo-implications. PROOF. Theorem 4.8 says that in an associative, commutative Hopf

algebra, if a primitive element is decomposable then it is a pt" power of another element. The Hopf algebra E(r) is associative and commutative and since x is primitive, yi = d is primitive and in an odd dimension; hence cannot be a pth power, and thus is not decomposable. By Lemma 4.3, there exists a primitive element ye E (r such that j(y) * 0 and hence x(d(r)y) (d(rjx)(Y) = (0j)(y) * 0. Since y is primitive, x = d(r)y is primitive also, hence of co-implications by Theorem 6.1. Hence x is of co-implications.

THEOREM 6.13. Let X be an arcwise connected H-space with Hi(X) finitely generated for all i and Hi(X; Z,) # 0 for at most a finite number of integers i. Then P(E(n) c image i(r) or in other words, even dimensional primitive elements in the cohomology Bockstein spectral sequence are the images of integral cocycles. For r = 1, this becomes P(H2n(X; Zr)) c image j..

PROOF. If x e P(E2,), and d(r)X * 0, then x has co-implications by Theorem 6.12 and thus Hi(X; Z,) * 0 for infinitely many i, contradicting the hypothesis. Hence x is a cycle for every d(r). Applying Theorem 3.11, we obtain the result.

REMARK. Suppose, in the hypothesis of Theorem 6.1, we change the condition x = d(r)y to d(r)x = y. Then the theorem is false without addi- tional hypotheses. For example in H8(F4; Z3) there is a primitive element (see [3, Theorem 19.26]). Similarly this example shows Theorem 6.12 would no longer be true if the condition d(r)x = y # 0 were replaced by x = d(r)z.

7. The Poincare duality theorem In this section we will prove that the Poincare duality theorem holds


for H-spaces with finitely generated homology, without assuming any sort of manifold structure.

First we quote from a theorem of Borel [1, Theorem 6.1], or [14]. THEOREM. (Borel). Let A be an associative, commutative Hopf algebra

over a perfect field k. Then A = ?, Ai as an algebra (not necessarily as a co-algebra), where the Ai are Hopf algebras with one generator. Thus if a, generates Ai, the set of monomials {ag' ... agn} forms an additive basis for A, where 0 ? q, < height of a,, where height of a, = smallest integer t > 0 such that at = 0 (a 1).

As usual p is a prime. Define the p-dimension of a space X to be the largest integer m such

that Hm(X; Z,) * 0. Similarly the rational dimension of X is the largest m such that Hm(X; Q) * 0, where Q = rational numbers.

THEOREM 7.1. If Xis an arcwise connected H-space with H,(X) finitely generated for each i, and if the p-dimension of X is finite, then the p-dimension of X equals the rational dimension of X, Hm(X; Z,) is one- dimensional, and Hm(X; Z,) = j*(Hm(X; Z)), (j: Z - Z,).

PROOF. By Borel's theorem, since H*(X; Z,) is an associative commutative Hopf algebra over Z,, there is a set of elements a,, a2, ..., an, (only a finite number because H*(X; Z,) is finite dimensional), a, e H*(X; Z,), such that the set of monomials {a1l ... agn} where 0 < q. < height of ai, forms an additive basis for H*(X; Zr). Since H*(X; Z,) is finite dimensional, we have that height of a, < co, for all i. Then there is a unique element of this form of highest dimension m, namely when each qi is maximal.

Now suppose that w1(X) = 0. Hence H1(X) = w1(X) = 0, and therefore H1(X;Z,) = 0. Hence dim a, > 2 for all i, so that if some qi is less than its maximum possible value, then dim (a'l ... agn) < m - 1. Thus Hm-1(X; Z,) = 0 and thus Em = Em) = Z,. Hence in homology also E(') = Em' = Z, and the result follows.

If X is not simply connected we take X = universal covering H-space of X. Then the results of [7] show that X satisfies the hypothesis of the theorem, and p-dimension of X = (p-dimension of X) - r, while rational dimension of X = (rational dimension of X) -r, where r = rank of w1(X) = dim H1(X; Q) (cf., the proof of theorem 6.10). Hence, using the result for X, we have it for X.

Hence E,') = E.-, and all elements of E(r) are cycles under d(r), so that there is no p-torsion in Hmi,(X). Hence we have:

COROLLARY 7.2. If X is an arcwise connected H-space with Hi(X) finitely generated for all i, Hm(X) = 0 and H3(X) = 0 for j > m, then Hm(X) = Z and H.__(X) is free.

48 WILLIAM BROWDER

COROLLARY 7.3. If Mis an arcwise connected, compact manifold which is an H-space, then M is orientable.

Corollary 7.3 also follows from an argument using a covering space of M.. Let A be the diagonal map of space X, A: X-) X x X, A(x) = (x, x),

so that A induces a map of C*(X) into C*(X) 0 C*(X), again denoted by A. Let C* = C*(X), C* C*(X), etc.

We recall the definition of cap product:

DEFINITION. Let cq e Cq en e Cn and let ten = kak 0 ak. Define cap product -^: Cq ( ) Cnq by ( 1 ) Eqe = (cq (g0 1)(Aen)-k (cq(ak))a.

If one defines a new differential 8' in C*, 8(: Cq Cq+1 by 8(' ( - )q?1(3

then the homology groups of 8' are identical with those of 8.

LEMMA 7.4. With the differential 8'on C*, and grading C* in reverse: (i.e., 'Cq = C-%, is a chain map of 'C* (D C* >) C*

PROOF.

-(9(c<( 0 en)) = ((_1)q11(8cq) 0 en) + -((-1)qc (0 &en)

== (-1)q+l E (8c0(ak))ap + (-_)q E (Cq(8aJ)af

+ ( 1) E (-l)dimak(cq (ak))ak'

= E c<(aj)Ja= 8(0en) Thus -- induces cap products in homology.

LEMMA 7.5. Let 8en = 0. Then Fen: 'Cam Cn__ is a chain map (8' orn 'Cq).

PROOF. 0 = Eq^e, = 8Cq en + (-1)q9(cq-en) so that (8'c<1)^en =i(c<q-en)- q.e.d.

Let f: A - B be a map of free chain complexes. PROPOSITION 7.6. f*: H(A) - H(B) is an isomorphism if and only if

fG: H(A 0 G) - H(B 0 G) is an isomorphism for all abelian groups G. PROOF. The second statement implies the first trivially by taking G = Z. If A* is an isomorphism, we look at the induced maps of the Universal

Coefficient Theorem [10, p. 161] sequences for H(A (0 G) and H(B (0 G) 0 - Hn(A) G > Hn(A 0 G) > Tor (Hn-1(A) G) > 0

f*81 a|IfG Tor(f*, 1){

O - > Hn(B) (JG > Hn(B 0g G) > Tor (Hn-,(B)~ G) > 0

But f* an isomorphism implies f * 0 1 and Tor (f*, 1) are isomorphisms. Hence fG is an isomorphism by the Five Lemma [10, p. 16].

PROPOSITION 7.7. Let f: A - B be a map of free chain complexes with


Hi(A), Hi(B) finitely generated for each i. Then f,: H(A) H(B) is an isomorphism if and only if fp: H(A 0 ZP) H(B 0 Z.) is an isomorphism. for every prime p.

PROOF. f * an isomorphism implies f, an isomorphism by Proposition 7.6. Define the complex C, called the "mapping cone" of f (see[1O, p. 155])

by Cn = An-l + Bn with d': Can Cn- defined by d'(a, b) =(-da, db +f(a)). Then g(a, b) = a is a map of degree -1 of C into A, (dg = -gd') and i(b) = (0, b) is a chain map of B into C. It follows that

(*) 0 > B C > A > 0 is an exact sequence of free chain complexes. If we take the homology sequence of (*) with coefficients in an abelian group G, we get an exact sequence:

(**) * * *

> Hn(B 0 G) -*>Hn(C (8& G)

> Hn- H (A (0 G) >Hn-1(B 0G) > ...

and one can verify directly that d* - fG. Setting G = Z we find that H,(C) is finitely generated since H,(B) and Hi-1(A) are. Setting G = Z, in (**) we find that since f, is an isomorphism H,(C 0 Zv) = 0 for all i, all p. But H,(C 0 Zv) = H,(C) (0 Zv + Tor (... *) so that H,(C) (0 Zv = 0, all i, all p. Then this implies that H,(C) = 0 since H.(C) is finitely generated. Again setting G = Z in (**) we have d* f* and hence f*: H(A) H(B) is an isomorphism.

LEMMA 7.8. Let X be an arcwise connected H-space with H,(X) finite dimensional for all i, and suppose Hm(X; Zv) * 0 and Hk(X; Zv) = 0, k > m. There exists a chain 4a e Cm(X), such that &4a = 0 and j*{ a} generates Hm(X; Z.). Then -mu: Hq(X; ZV) Hm-q(X; Z.) is an isomorphism for all q.

PROOF. By Theorem 7.1, Hm(X; Zv) = j*(Hm(X; Z)) so that there is a 4a e Cm(X) such that &4a = 0 and j*{4a} generates Hm(X; Zv).

By Borel's theorem we may find a set of generators {a1, *., an} for H*(X; Z.) so that {a"1 .. a 0$n, 0 _ qi <h h = (height a,) -1, form an additive basis for H*(X; Z.). For each monomial a = a" ... aqn, assign another monomial f (a) = a" ... an, where ti = hi-qi. Then if {b1, * * , bk} is a monomial basis for Hq(X; ZV), {If (b1), * - I, f (bj)} is a monomial basis for Hm Z(X;.Z). Also a .f(a) = al... a" Further we show that bi f(bj) = 0 if i #j. For some exponent ts, (bj = a" ... a**n bi = al1 * ... arn) we have ts < rs and thus f(bj) = a1`-t1 ... aAntn so that b f(bj) ) a 1-t1+r1 ..s. aA-ts+rs ... aAn-tn+rn = 0 since aAS-ts+rs = 0 because

__ a1 a h t aT p0iring es becsau set+r>s hs + 1 = height of as. Thus the product pairing establishes an isomorphism-

50 WILLIAM BROWDER

of H9(X; Z,) with (Hr-9(X; Z,))*. Hence, if x e H9(X; Zr), x # 0, there is ye Hmrn(X; Zp) such that xmy = a, * an. Now aj ... an (j*{4a}) * 0, so that (x*y)(4a) = (x*y)(j* {4}) * 0.

But y(x-ji) = y((x 0 1)(zAp)) = (x 0 y)(?Aje) = (xy)(4a) # 0. Hence no element y e H*(X; Z,) is orthogonal to image Emu, so mu is onto. Also no element x e H*(X; Z,) goes to zero under -qu so -qu is one to one. Hence -mu is an isomorphism H9(X; Z,) with Hm-q(X; Zr).

THEOREM 7.9. (Poincare Duality Theorem). Let X be an arcwise connected H-space with H,(X) finitely generated for all i and suppose Hm(X) # O Hj(X) = 0 for j > m. Then Hm(X) = Z and if e is a generator of Hm(X), then -At: H9(X; G) Hmgq(X; G) is an isomorphism for all q, for any abelian group G.

PROOF. By the definition of -I it suffices to prove the theorem for mu where ,u is any chain representing &.

By Corollary 7.2, Hm(X) = Z. By Lemma 7.8, mu is an isomorphism for G = ZP for all q, all prime p. By Proposition 7.7, mu is then an isomorphism for G = Z and hence by Proposition 7.6, -mu is an isomorphism for any abelian group G. q.e.d.

CORNELL UNIVERSITY AND UNIVERSITY OF CHICAGO

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torsion in h-spaces - university of edinburgh

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