ON HOMOLOGICAL RATE OF GROWTH AND THE HOMOTOPY TYPE OF ΩBG_{p}^{∧}

RAN LEVI

April 18 1995

Abstract. Let G be a finite p-perfect group. We show that the mod-p homology of ΩBG_{p}^{∧} grows either polynomially or semi-exponentially. A conjecture due to F. Cohen states that ΩBG_{p}^{∧} for such groups G is spherically resolvable of finite weight. We show that any space X, which satisfies the conclusion of Cohen’s conjecture has the property that its homology grows at most hyper-polynomially of finite degree. Thus we conclude that if a group G satisfies the Cohen conjecture then the homology of ΩBG_{p}^{∧} grows polynomially. This enable us to produce counter examples to the conjecture. We study some further homotopy properties of our examples. We also show that H_{∗}(ΩBG_{p}^{∧}) is a finitely generated, Lie nilpotent algebra provided it grows polynomially.

1. Introduction

Let p be a prime number. Recall that a group G is said to be p- perfect if the first homology group of G with coefficients in the prime field 𝔽_{p} vanishes. This paper is motivated by a conjecture of F. Cohen [4], which suggests that the loop space ΩBG_{p}^{∧} is spherically resolvable of finite weight for every finite p-perfect group G, where (−)_{p}^{∧} denotes the Bousfield-Kan p-completion functor.

For every graded vector space M let P_{M}(t) denote the Poincaré series for M, in which the coefficient of t^{d} we shall denote by θ_{M}(d). We let F_{M}(d) denote the sum ∑_{i=0}^{d} θ_{M}(i). The function F_{M}(d) shall be called the growth function of M. Throughout this paper all spaces are assumed to have the homotopy type of a CW complex and to be connected and p-complete for some fixed prime p. By H_{∗}(−) we shall denote mod-p homology. If X is a space then we write F_{X}(d) when we mean F_{M}(d) for M = H_{∗}(X).

1991 Mathematics Subject Classification. Primary 55R35, Secondary 55R40, 55Q52.

The author is supported by a DFG grant

Consider the mod-p homology of a space X, which is spherically resolvable of finite weight. A crude upper bound for the function F_{X}(d) can be obtained by looking at the growth functions F_{n,k}(d) = F_{M}(d) for M = H_{∗}(Ω^{n}S^{n+k}). On the other hand for spaces of the form ΩBG_{p}^{∧}, it turns out that being spherically resolvable of finite weight imposes a severe restriction on the homological rate of growth, namely in that case H_{∗}(ΩBG_{p}^{∧}) must grow polynomially. We start with a precise account of these ideas.

Definition 1.1. A graded vector space M is said to grow hyper-polynomially of degree n if there exists an integer m and a positive constant r such that

lim_{d→∞} \frac{F_{M}(d)}{d^{r(log(d))^{m}}} = 0.

and n + 1 is the least such m.

The vector space M is said to grow semi-exponentially if there exists a real number λ > 1 such that F_{M}(d) ≥ λ^{d^{α}} for some constant 0 < α < 1 and sufficiently large values of d

Our first observation is a rather straight forward consequence of a theorem due to Y. Felix, S. Halperin and J. C.Thomas [7], which states that for finite complexes X there is a dichotomy in the possible growth rates of H_{∗}(ΩX). We use this result to show that a similar dichotomy holds for spaces of the form BG_{p}^{∧}.

Theorem 1.2. Let G be a finite p-perfect group. Then H_{∗}(ΩBG_{p}^{∧}) grows either polynomially or semi-exponentially.

On the other hand, motivated by the Cohen conjecture we study the rate of growth of H_{∗}(Ω^{n}S^{n+k}), using the well known structure theorems for these spaces.

Theorem 1.3. For each pair of positive integers n and k H_{∗}(Ω^{n}S^{n+k}) grows hyperpolynomially of finite degree depending on n, k and the prime p. Thus if X is spherically resolvable of finite weight then H_{∗}(X) grows hyper-polynomially of finite degree.

It is easy to see that if M grows hyper-polynomially of finite degree and N grows semi-exponentially then lim_{d→∞} \frac{F_{M} (d)}{F_{N} (d)} = 0. Thus an immediate corollary of Theorem 1.3 is that if H_{∗}(X) grows semi-exponentially then X is not spherically resolvable of finite weight. Restricting our attention to spaces of the form ΩBG_{p}^{∧}, a direct consequence is the following Corollary 1.4. If G is a finite p-perfect group, which satisfies the Cohen conjecture then H_{∗}(ΩBG_{p}^{∧}) grows polynomially.

A counter example to Cohen’s conjecture was recently discovered [10], making use of the ideas presented here. A different family of counter examples is given in the sequel.

Theorem 1.5. For any prime p ≥ 5 and a positive integer r there exists a finite p-perfect group E(p^{r}) of order 3p^{2r} such that H_{∗}(ΩBE(p^{r})_{p}^{∧}) grows exponentially. Thus ΩBE(p)_{p}^{∧} does not satisfy Cohen’s conjecture. Moreover the loop space on a 6-dimensional Moore space ΩP^{6}(p^{r}) is a retract of ΩBE(p^{r})_{p}^{∧}.

The fact that one can obtain the loop space on a Moore space as a retract of a space of the form ΩBG_{p}^{∧} for G finite and p-perfect is mildly surprising in view of the non-existence of essential maps from BG_{p}^{∧} into a finite complex. This may raise several questions about the generality of such phenomena.

The proof of Theorem 1.5 allows an implicit computation of H_{∗}(ΩBE(p^{r})_{p}^{∧}).

Theorem 1.6. Let T denote the tensor algebra generate by a_{1}, b_{2}, b′_{2}, c_{4}, c′_{4} and d_{5}, where degrees are given by subscripts. Then there is 6-dimensional primitive element γ ∈ T and an isomorphism of T/T γT -modules

H_{∗}(ΩBE(p^{r}))_{p}^{∧}) ≅ T/T γT ⊗ H_{∗}(SU(3)).

In particular H_{∗}(ΩBE(p^{r}))_{p}^{∧}) is finitely generated as an algebra.

Considerably many positive examples to the Cohen conjecture are known [9]. In each and every one of the examples known to the author of groups G, for which ΩBG_{p}^{∧} is spherically resolvable of finite weight it turns out that H^{∗}(BG) has a very particular structure. Indeed, there is a polynomial subalgebra P of H^{∗}(BG) such that H^{∗}(BG)//P is an exterior algebra. In view of our results here, one might be tempted to speculate that the Cohen conjecture for a finite p-perfect group G holds if and only if H^{∗}(BG) admits a filtration with an associated graded, which is a symmetric algebra.

If one is willing to restrict attention to groups G for which the loop space homology of BG_{p}^{∧} grows polynomially, then one can infer from structure theorems of Felix, Halperin and Thomas, concerning the loop space homology of spaces of finite category to conclude similar results for spaces of the form ΩBG_{p}^{∧}.

Theorem 1.7. Suppose that G is a finite p-perfect group such that H_{∗}(ΩBG_{p}^{∧}) grows polynomially. Then H_{∗}(ΩBG_{p}^{∧}) is a finitely generated algebra for which the associated Lie algebra is nilpotent.

It is still far from being clear what should be a necessary and sufficient condition on G, which would imply either polynomial growth of H_{∗}(ΩBG_{p}^{∧}) or, more strongly, that ΩBG_{p}^{∧} satisfies the Cohen conjecture.

The author takes the pleasure to thank the University of Heidelberg for its support. We are also indebted to F. Cohen, J. Harper and H. W. Henn for many helpful conversations.

2. rate of growth and Resolution by Fibrations

The mod-p homology algebras of iterated loop spaces on spheres are symmetric algebras on certain sets of generators. Thus we study the rate of growth of such algebras.

For any positively graded vector space M over a field κ, let T (M) denote the free associative algebra generated by M. Let I denote the two-sided ideal generated by elements xy − yx for all x, y ∈ M. DefineP (M) = T (M)/I. The symmetric algebra on a graded vector space M (concentrated in even degrees if char(κ) is odd) is the graded commutative analogue of algebras of the form P (M). For our purpose it is more convenient to consider algebras of this form rather then symmetric algebras.

For a graded vector space M, let P_{M}(t), θ_{M}(d) and F_{M}(d) denote the Poincaré series, the coefficient of t^{d} and the growth function for M respectively. A function f(d) from the integers to the real numbers is said to be a complexity upper bound for M if the sequence \frac{F_{M} (d)}{d^{f(d)}} is bounded.

Lemma 2.1. Let M be a positively graded vector space. Then F_{M}(d) is a complexity upper bound for P (M).

Proof. Let K(d, n) denote the number of monomials in n variables of degree d. Let x_{1}, x_{2}, ⋯, x_{n} be a basis for ∑_{i=1}^{d} M_{i}. Then θ_{P (M)}(d) ≤ K(d, F_{M}(d)) for if |x_{i}| = 1 for all 1 ≤ i ≤ n, then equality holds. Since θ_{P (M)}(d) increases with d,

F_{P (M)}(d) ≤ d · θ_{P (M)}(d) ≤ d · K(d, F_{M}(d)).

One easily verifies the following formula

K(d, n) = \binom{d + n − 1}{n − 1} = \frac{1}{(n − 1)!}(d^{n−1} + q(d)),

where q(d) is a polynomial in d of degree strictly less than n − 1. Thus

F_{P (M)}(d) ≤ \frac{1}{(F_{M}(d) − 1)!}(d^{F_{M} (d)} + q(d))

and so the sequence (F_{P (M)}(d))/(d^{F_{M} (d)}) is bounded.

Corollary 2.2. Let M be a positively graded vector space. Then F_{M}(d) is a complexity upper bound for S(M). Thus if lim_{d→∞} \frac{F_{M} (d)}{(logd)^{n}} = 0, for some positive integer n then S(M) grows hyper-polynomially of degree at most n.

Next we consider the concept of resolutions by fibrations. Let C be a family of connected spaces. A space X is said to be resolvable by fibrations over C if there exists a tower of principal fibrations

⋯ ⟶ X_{n} ⟶ X_{n−1} ⟶ ⋯ ⟶ X_{2} ⟶ X_{1} = X

such that

(1) For each i ≥ 1 there exists B_{i} ∈ C such that the fibration, X_{i+1} ⟶ X_{i}, is induced from the path loop fibration over B_{i} via a map ρ_{i} : X_{i} ⟶ B_{i}.

(2) lim_{\overleftarrow{i}} X_{i} is weakly contractible.

In particular we say that X is resolvable by fibrations over C of weight ≤ n if there exists a tower, as above, such that X_{n} is homotopy equivalent to an object of C. A sequence, as above, is said to be a resolution for X over C. The respective spaces B_{i} are called the resolving spaces for the given resolution.

Let S be the family of all spheres and iterated loop spaces Ω^{n}S^{n+k} for k ≥ 1. Spaces X, which are resolvable by fibrations over S of finite weight are said to be spherically resolvable of finite weight.

It is immediate, by considering the Serre spectral sequence, that if a space X admits a resolution by fibrations over spaces B_{1}, ⋯ B_{n}, then the growth function for H_{∗}(X) is bounded from above by the growth function for

H_{∗}(B_{1}) ⊗ H_{∗}(B_{2}) ⊗ ⋯ H_{∗}(B_{n}).

Indeed, in the case where the Serre spectral sequence collapses for each one of the fibrations involved, the two functions are equal.

3. the dichotomy

Let G be a finite group. Then there is a faithful representation

ρ : G ⟶ SU(n)

for some n. By classifying ρ we obtain a fibration

SU(n)/ρ(G) ⟶ BG ⟶ BSU(n). (3.1)

The fibre in 3.1 will usually be denoted by SU(n)/G. Since the base space in 3.1 is simply-connected, the p-completion functor (−)_{p}^{∧} preserves the fibration and we get a fibration

(SU(n)/G)_{p}^{∧} ⟶ BG_{p}^{∧} ⟶ BSU(n)_{p}^{∧}. (3.2)

The fibration 3.2 plays an important role in our study. In particular it enables us to use the dichotomy theorem of Felix, Halperin and Thomas [7] to conclude our Theorem 1.2. We proceed by its proof.

Assume that G is p-perfect and let a faithful representation of G in SU(n) be given. By pulling back 3.2 we obtain a sequence of fibrations

ΩSU(n)_{p}^{∧} ⟶ Ω(SU(n)/G)_{p}^{∧} ⟶ ΩBG_{p}^{∧} ⟶ SU(n)_{p}^{∧}.

Considering the fact that SU(n) is finite and that the homology of ΩSU(n) grows polynomially, we conclude that

(1) H_{∗}(ΩBG_{p}^{∧}) grows polynomially if and only if so does H_{∗}(Ω(SU(n)/G)_{p}^{∧}).

(2) H_{∗}(ΩBG_{p}^{∧}) grows semi-exponentially if and only if so does H_{∗}(Ω(SU(n)/G)_{p}^{∧}). Now, the dichotomy theorem of [7] states that if X is a space of finite category

(in particular a finite complex) then H_{∗}(ΩX) grows either polynomially or semiexponentially. Since (SU(n)/G)_{p}^{∧} has the homotopy type of a finite complex, it follows that H_{∗}(ΩBG_{p}^{∧}) grows either polynomially or semi-exponentially. This completes the proof of Theorem 1.2.

4. the growth of H_{∗}(Ω^{n}S^{n+k})

In this section we prove Theorem 1.3. The proof follows by a direct calculation using the well known structure theorems for the mod-p homology of iterated loop spaces on spheres. We carry out an explicit calculation at the prime 2 and remark that the proof for odd primes follows by similar methods. In fact for any prime p one can show that the algebra P (M) grows hyper-polynomially of finite degree, where M is such that H_{∗}(Ω^{n}S^{n+k}) is isomorphic to the symmetric algebra on M.

We start by reviewing the structure theorem for H_{∗}(Ω^{n}S^{n+k}) at the prime 2.

consider sequences

I = (i_{1}, i_{2} ⋯, i_{s}),

With each such sequence I, we associate a length l(I) = s, a degree d(I) = ∑_{j=1}^{s} i_{j} and an excess e(I) = i_{1} − ∑_{j=2}^{s} i_{j}. A sequence I is said to be admissible if 2i_{j} ≥ i_{j−1} for 2 ≤ j ≤ k. The Dyer-Lashof operation Q^{I} corresponding to a sequence I, as above is given by

Q^{I} = Q^{i_{1}}Q^{i_{2}} ⋯ Q^{i_{s}}.

Recall [5] that H_{∗}(Ω^{n}S^{n+k}) is the polynomial algebra on Dyer-Lashof operations Q^{I} applied to the fundamental class g_{k} ∈ H_{k}(Ω^{n}S^{n+k}). The sequences I are required to satisfy the following three conditions

(1) I is admissible,

(2) e(I) > k and

(3) i_{s} < n + k.

We call a sequence I satisfying (1) through (3) above for given values of n and k, an (n, k)-sequence. One immediately verifies that if I = (i_{1}, i_{2}, ⋯ i_{s}) is an (n, k)- sequence, then so is the sequence (i_{2}, ⋯ i_{s}).

Lemma 4.1. Any (n, k)-sequence I = (i_{1}, i_{2}, ⋯ i_{s}) has the form

I = (2^{s−1}k + 2^{s−2}j_{s} + ⋯ + j_{2} + j_{1}, ⋯, 2k + j_{s} + j_{s−1}, k + j_{s}),

for some 1 ≤ j_{1} ≤ j_{2} ≤ ⋯ ≤ j_{s} ≤ n − 1.

Proof. For s = 1 one immediately verifies the lemma. Thus assume the lemma for l(I) = s and prove for s + 1. Indeed write

I = (i_{1}, 2^{s−1}k + 2^{s−2}j_{s+1} + ⋯ + j_{3} + j_{2}, ⋯, k + j_{s+1}).

Then the integer i_{1} has to satisfy,

i_{1} ≤ 2i_{2} = 2^{s}k + 2^{s−1}j_{s+1} + ⋯ + 2j_{3} + 2j_{2},

for the sequence to be admissible and the excess e(I) must be bigger than k. In calculating the sum ∑_{j=2}^{s+1} i_{j}, we get

∑_{j=2}^{s+1} i_{j} = (2^{s} − 1)k + 2^{s−1}j_{s+1} + ⋯ + 2j_{3} + j_{2}.

The excess condition implies

i_{1} ≥ k + (∑_{j=2}^{s+1} i_{j}) + 1 = 2^{s}k + 2^{s−1}j_{s+1} + ⋯ + 2j_{3} + j_{2} + 1.

On the other hand admissibility gives

i_{1} ≤ 2^{s}k + 2^{s−1}j_{s+1} + ⋯ + 2j_{3} + 2j_{2}.

Thus one can write

i_{1} = 2^{s}k + 2^{s−1}j_{s+1} + ⋯ + 2j_{3} + j_{2} + j_{1},

for some 1 ≤ j_{1} ≤ j_{2}. This completes the proof of the lemma.

Corollary 4.2. The number P_{n,k}(s) of (n, k)-sequences of length s is precisely the number of all sequence j_{1}, ⋯ j_{s} with

1 ≤ j_{1} ≤ j_{2} ≤ ⋯ ≤ j_{s} ≤ n − 1.

In particular this number is independent of k.

We next compute P_{n,k}(s) for large values of s.

Proposition 4.3. Let s ≥ n − 1. Then

P_{n,k}(s) = ∑_{i=0}^{n−2} \binom{s − 1}{i}\binom{n − 1}{i + 1}

Proof. Each sequence J = (j_{1}, ⋯, j_{s}) is determined by two parameters, which can be described as follows.

(1) How many different labels among the numbers 1, ⋯ n − 1 are in it and

(2) Given a fixed number of labels, how many ways are there to distribute them among the indices 1, ⋯ s.

There is precisely one way to distribute one label; \binom{s−1}{1} ways to distribute two

labels; \binom{s−1}{2} ways to distribute three labels etc. Respectively, there are \binom{n−1}{1} choices

of one label; \binom{n−1}{2} choices of two labels etc. Multiplying out and summing we get the required formula.

Define D_{n,k}(s) = ∑_{i=0}^{s} P_{n,k}(i). Then D_{n,k}(s) is the function which counts how many generators exist in H_{∗}(Ω^{n}S^{n+k}) with the respective Dyer-Lashof operation of length at most s. Furthermore, write

D_{n,k}(s) = ∑_{i=0}^{n−1} P_{n,k}(i) + ∑_{i=n}^{s} P_{n,k}(, i).

Let A_{n,k} denote the first summand and notice that it is independent of s.

A sequence of length s, I = (i_{1}, ⋯, i_{s}) has a maximal dimension if the corresponding sequence (j_{1}, ⋯, j_{s}) satisfies j_{r} = n − 1 for all r. Let I_{s}^{M} denote that sequence. Similarly, let I_{s}^{m} denote the sequence of length s and minimal dimension. Notice that I_{s}^{m} is obtained by taking j_{r} = 1 for all 1 ≤ r ≤ s. If j_{r} = j_{t} for all 1 ≤ r, t ≤ s, then by Lemma 4.1, we get for the corresponding sequence I, d(I) = (2^{s} −1)(k +j_{r}). Thus in H_{∗}(Ω^{n}S^{n+k}) we have

|Q^{I_{s}^{M}} g_{k}| = (2^{s} − 1)(k + n − 1) + k and |Q^{I_{s}^{m}}g_{k}| = (2^{s} − 1)(k + 1) + k.

For any positive integer r, one has the formula

|Q^{I_{s+r}^{m}}g_{k}| − |Q^{I_{s}^{M}} g_{k}| = (2^{s}(2^{r} + (2^{r} − 1)k − n + 1) + n − 2).

Hence if r is sufficiently large there are no generators corresponding to sequences of length at least s + r in dimension less than (2^{s} − 1)(k + n − 1) + k. In particular the number of algebra generators appearing in H_{∗}(Ω^{n}S^{n+k}) in dimensions less than or equal to (2^{s} − 1)(k + n − 1) + k is bounded from above by D_{n,k}(s + r) for all s and a fixed r suitably chosen.

Proposition 4.4. For all positive integers n and r

lim_{s→∞} \frac{D_{n,k}(s + r)}{s^{n}} = 0.

Proof. For large values of s we have the equality

D_{n,k}(s) = A_{n,k} + ∑_{j=n}^{s} P_{n,k}(j) = A_{n,k} + ∑_{j=n}^{s} ∑_{i=0}^{n−2} \binom{j − 1}{i}\binom{n − 1}{i + 1}.

Put B_{i} = \binom{n−1}{i+1} and T_{i}(s) = ∑_{j=n}^{s} \binom{j−1}{i}. Then the equality above reads

D_{n,k}(s) = A_{n,k} + ∑_{i=0}^{n−2} B_{i}T_{i}(s).

Notice that the coefficients B_{i} are independent of s. We prove the proposition by showing that T_{i}(s) is bounded above by a polynomial in s with positive coefficients of degree at most i + 1.

Indeed it is immediate that \binom{j−1}{i} is a polynomial in j of degree i. Thus for sufficiently large s,

T_{i}(s) ≤ s · \binom{s − 1}{i},

which is a polynomial in s of degree i + 1. This completes the proof.

We are now ready to prove Theorem 1.3. Let Δ_{n,k}(d) denote the number of polynomial generators, which appear in H_{∗}(Ω^{n}S^{n+k}) up to dimension d. Then by the argument above there exists a positive integer r such that D_{n,k}(s + r) ≥ Δ_{n,k}(α(s)), where α(s) = (2^{s} − 1)(k + n − 1) + k. One verifies with no difficulty that for sufficiently large values of s, α(s) ≥ 2^{s}. Since Δ_{n,k}(s) is an increasing function, we get D_{n,k}(s + r) ≥ Δ_{n,k}(2^{s}). Now setting d = 2^{s}, we have s = log_{2}d and the theorem follows at once from Proposition 4.4 and Corollary 2.2.

5. a counter example to the Cohen conjecture

In this section we prove theorem 1.5. Thus we construct a family of finite pperfect groups E(p^{r}), one for each odd prime p ≥ 5 and a positive integer r, such that ΩBE(p)_{p}^{∧} grows exponentially. Our examples here are similar in spirit to those exhibited in [10].

The group ℤ/3ℤ admits a faithful representation in GL_{2}(ℤ), given as follows. Let σ denote a generator of ℤ/3ℤ. Let σ operate on a free abelian group on two generators x̄ and ȳ by sending x̄ to ȳ and ȳ to −(x̄+ȳ). Let ρ denote the resulting representation. For any prime p and a positive integer r there is a faithful representation of ℤ/3ℤ in GL_{2}(ℤ/p^{r}ℤ), obtained by reducing ρ. Thus ℤ/3ℤ operates on a free ℤ/p^{r}ℤ-module T_{r} on two generators x and y via this representation. Define E(p^{r}) to be the semi-direct product of T_{r} by ℤ/3ℤ with respect to this operation.

Notice that for p = 3, we obtain 3-groups, which are out of our context. We present a cohomology calculation for E(p^{r}) assuming p ≥ 5. The case p = 2 is particularly interesting. Indeed E(2) ≅ A_{4}, the alternating group on 4 letters, for which the 1-connected cover (Ω(BA_{4})_{2}^{∧})⟨1⟩ is spherically resolvable of length 2 [9]. Thus it is not a counter example. Nevertheless, E(2^{r}) for r > 1 behaves very similarly to the odd prime case, at least in terms of group cohomology. We were not able to decide whether the loop space homology of BE(2^{r})_{2}^{∧} grows exponentially for r > 1.

For any prime p and an integer r, the group E(p^{r}) is given by generators and relations as follows.

E(p^{r}) = ⟨x, y, σ | x^{p^{r}} = y^{p^{r}} = σ^{3} = [x, y] = 1; x^{σ} = y; y^{σ} = x^{−1}y^{−1}⟩.

Define a faithful representation ρ : E(p^{r}) ⟶ SU(3) by sending x to the diagonal matrix diag(ζ, ζ^{−1}, 1), y to diag(ζ^{−1}, 1, ζ) and σ to the permutation matrix, whose i-th row is the unit vector e_{i+1}, i = 1, 2 and whose third row is e_{1}. One immediately verifies that the relations are satisfied.

Proposition 5.1. For each prime p ≥ 5 there is an isomorphism of algebras

H^{∗}(SU(3)/E(p^{r})) ≅ P [a_{2}, b_{3}, b′_{3}, c_{5}, c′_{5}, d_{6}]/R,

where degrees are given by subscripts and the set of relations R consists of the relation ad = cb = c′b′ together with setting every other product being equal to zero. The action of the Steenrod algebra on H^{∗}(SU(3)/E(p^{r})) is trivial unless r = 1, in which case there are non trivial Bockstein operators. In fact for every r the Bockstein spectral sequence is determined by the requirement that β_{r}a = b and β_{r}c = d.

Proof. Let T_{r} ≅ ℤ/p^{r}ℤ ⊕ ℤ/p^{r}ℤ denote the Sylow p-subgroup of E(p^{r}). It is easy to check that the map induced on mod-p cohomology by the composite

BT_{r} ⟶ BE(p^{r}) ⟶ BSU(3)

takes the Chern classes c_{2} and c_{3} to the classes u^{2}+uv+v^{2} and u^{2}v+uv^{2} respectively, where

H^{∗}(BT_{r}) ≅ P [u_{2}, v_{2}] ⊗ E[x_{1}, y_{1}],

with β_{r}x = u and β_{r}y = v. Obviously these two classes are invariant under the ℤ/3ℤ action of H^{∗}(BT_{r}). By the big collapse theorem of L. Smith [12], it follows that

H^{∗}(SU(3)/E(p^{r})) ≅ H^{∗}(BE(p^{r}))/(w_{4}, w_{6}),

where w_{2j} denotes the image of c_{j}.

One proceeds with computing the ring of invariants (H^{∗}(BT_{r}))^{ℤ/3ℤ} modulo the ideal generated by u^{2} + uv + v^{2} and u^{2}v + uv^{2}. Let res : H^{∗}(BE(p^{r})) ⟶ H^{∗}(BT_{r}) denote the restriction. We conclude the proof by listing the images of the generators and leave it to the reader to verify that all the relations as well as the action of the Steenrod algebra and Higher Bockstein operators hold modulo u^{2} + uv + v^{2} and u^{2}v + uv^{2}.

(1) res(a) = yx.

(2) res(b) = xv − yu.

(3) res(b′) = xu + yv + xv.

(4) res(c) = xu^{2} + 2xuv + yu^{2} − yv^{2}.

(5) res(c′) = xv^{2} + yu^{2} + 2(x + y)uv.

(6) res(d) = u^{3} + 3u^{2}v − v^{3}.

This cohomological information is in fact sufficient to conclude that H_{∗}(ΩBE(p^{r})_{p}^{∧}) grows exponentially if p ≥ 5 and hence that ΩBE(p^{r})_{p}^{∧} does not satisfy the Cohen conjecture. Theorem 1.5 however makes a stronger claim, namely that ΩP^{6}(p^{r}) is a retract of ΩBE(p^{r})_{p}^{∧}. Thus we proceed by proving this assertion. The calculation is of a spirit similar to the one presented in [10] but introduces further complications. The author take the pleasure to thank John Harper for explaining to him a theorem of I. James which allows one to obtain P^{6}(p^{r}) as a retract of (SU(3)/E(p^{r}))_{p}^{∧}. Throughout this section let X denote (SU(3)/E(p^{r}))_{p}^{∧}.

Before proceeding with analyzing the homotopy type of X, we observe that having P^{6}(p^{r}) as a retract of X implies that ΩP^{6}(p^{r}) is a retract of ΩBE(p^{r})_{p}^{∧}. Indeed for p ≥ 5, the space SU(3)_{p}^{∧} is homotopy equivalent to a product (S^{3})_{p}^{∧} × (S^{5})_{p}^{∧}. Since the fibre inclusion map SU(3)_{p}^{∧} ⟶ X induces the zero map in mod-p cohomology, so does the composite

f : SU(3)_{p}^{∧} ⟶ X ⟶ P^{6}(p^{r}).

This implies that f is null-homotopic when restricted the the wedge (S^{3})_{p}^{∧} ∨ (S^{5})_{p}^{∧}. Hence f factors through the 8-sphere (S^{8})_{p}^{∧}. But π_{8}P^{6}(p^{r}) = 0 for the primes under consideration and so f is null-homotopic. This gives a map ϕ : ΩBE(p^{r})_{p}^{∧} ⟶ ΩP^{6}(p^{r}), which is a left homotopy inverse to the composite

ΩP^{6}(p^{r}) ⟶ ΩX ⟶ ΩBE(p^{r})_{p}^{∧}.

In what follows we drop the p-completion symbol whenever possible but maintain the assumption that all spaces are p-complete. Let r be fixed and denote P^{n}(p^{r}) by P^{n}. The cohomological structure of X implies that

X ≃ (S^{3} ∨ P^{3} ∨ S^{5} ∨ S^{5}) ∪_{g+[p^{r}]} e^{6} ∪_{h} e^{8},

where [p^{r}] is a degree p^{r} map on one of the 5-cells and g represents a homotopy class, also denoted by g, in π_{5}(S^{3} ∨ P^{3}). Indeed the attaching of the 5-cells could be done via a multiple of a Whitehead product. But this would result in having the respective 5-class being a cup product of a 2 and a 3-class in cohomology with coefficients in ℤ/p^{r}ℤ, which is not the case. It is easy to observe that the Moore space P^{6} splits off from X if and only if g is null-homotopic.

Consider the complex

Z ≃ (S^{3} ∨ P^{3}) ∪_{g} e^{6} ∪_{h′} e^{8},

obtained from X by collapsing the two 5-cells to a single point. Write H̃^{∗}(Z) = 𝔽_{p}⟨a, b, b′, d, D⟩ with a single non-vanishing product ad = D. Let α denote the fundamental class in π_{2}(P^{3}).

Proposition 5.2. The cohomological structure of the complex Z implies that the Whitehead product [g, α] vanishes in π_{6}(S^{3} ∨ P^{3}), where g is the attaching map for the 6-cell in Z.

We shall later observe that if g′ ∈ π_{5}(S^{3} ∨ P^{3}) is any non-zero element then [α, g′] ≠ 0. Having done so, Proposition 5.2 gives that g is null-homotopic and the proof of Theorem 1.5 will thus be complete.

Denote S^{3} ∨ P^{3} by A and let Y = A ∪_{g} e^{6}. Let F be the homotopy fibre of the map Y ⟶^{π} S^{6}, obtained by collapsing A to a point. Then the natural map A ⟶^{i} Y factors through F and by turning maps into inclusions we obtain a triple (Y, F, A). Notice that π_{∗}(Y, F) ≅ π_{∗}(S^{6}) and so since p ≥ 5, π_{7}(Y, A) ≅ π_{7}(F, A), an isomorphism being induced by the inclusion (F, A) ⟶ (Y, A). It is easy to observe that H_{7}(F; ℤ) ≅ ℤ/p^{r}ℤ by inspection of the integral homology Serre spectral sequence associated to the fibration

ΩS^{6} ⟶ F ⟶ Y.

Since the pair (F, A) is 6-connected, π_{7}(F, A) ≅ H_{7}(F, A; ℤ) ≅ ℤ/p^{r}ℤ. On the other hand let σ_{6} : (D^{6}, S^{5}) ⟶ (Y, A) denote the attaching map, so that in the long exact sequence for the pair (Y, A), ∂(σ_{6}) = g. Then π_{7}(Y, A) contains the relative Whitehead product [σ_{6}, α], which is an element of order p^{r}. Hence we conclude that [σ_{6}, α] generates π_{7}(Y, A).

Lemma 5.3. Let h′ ∈ π_{7}(Y) be the attaching map such that Y ∪_{h′} e^{8} = Z. Then i_{∗}(h′) = [σ_{6}, α], where i_{∗} : π_{7}(Y) ⟶ π_{7}(Y, A) is the natural map.

Notice that the lemma implies Proposition 5.2, as in the exact sequence for the pair (Y, A), ∂[σ_{6}, α] = [g, α]. Before proving it, one more observation is needed. Let

c : Y ⟶ Y ∨ S^{6}

denote the cooperation map obtained by a middle pinch of the top cell (see [8] for a precise description). Let u : Y ⟶ Y ∨ S^{6} denote the inclusion and let w = [ι_{6}, α] ∈ π_{7}(Y ∨ S^{6}) be the Whitehead product.

Lemma 5.4. Let l ∈ π_{7}Y be any element. Then i_{∗}(l) = λ[σ_{6}, α], λ ∈ ℤ/p^{r}ℤ if and only if

c_{∗}(l) = u_{∗}(l) + λw.

Proof. This is just a slight variation on the proof of [8, Lemma 5.4], observing the fact that π_{7}(Y, A) is generated by the relative product [σ_{6}, α].

In order to complete the proof of Lemma 5.3 it suffices to show that c_{∗}(h′) = u_{∗}(h′) + w in π_{7}(Y ∨ S^{6}). Indeed let Z′, K_{u} and K_{λw} denote the complexes obtained from Y ∨S^{6} by attaching an 8-cell via maps of homotopy classes c_{∗}(h′), u_{∗}(h′) and λw respectively. Let x_{6} denote the class in H^{∗}(Y ∨ S^{6}) carried by S^{6}. Then in H^{∗}(K_{u}), ax_{6} = 0 and in H^{∗}(K_{λw}), ax_{6} = λ̄D, where λ̄ ∈ 𝔽_{p} denotes the class of λ mod p. On the other hand there is a natural map γ : Z ⟶ Z′, which in cohomology sends x_{6} to d and is the identity on all other classes. Thus by naturality we have in H^{∗}(Z′), ax_{6} = D. Since the attaching map for the 8-cell in Z′ is given by c_{∗}(h′) = u_{∗}(h′)+λw, we conclude that λ = 1 ([8, Thm. 4.1] provides justification for this argument). This completes the proof of Lemma 5.3 and hence of Proposition 5.2.

To complete the proof of Theorem 1.5 it remains to show that if g′ ∈ π_{5}(A) is any non-zero element then [α, g′] ≠ 0. Indeed let ι ∈ π_{3}S^{3} denote the fundamental class. By the Hilton-Milnor theorem

π_{5}(A) ≅ ℤ/p^{r}ℤ ⊕ π_{5}P^{3},

where the left summand is generated by the Whitehead product [α, [α, ι]]. Notice also that π_{5}(P^{3}) is cyclic. Thus write g′ = λ_{1}[α, [α, ι]] + λ_{2}δ for δ ∈ π_{5}P^{3} a generator. Then by naturality of the Whitehead product assuming [α, g′] = 0 we have

0 = [α, g′] = λ_{1}[α, [α, [α, ι]]] + λ_{2}[α, δ].

But [α, [α, [α, ι]]] ≠ 0 by the Hilton-Milnor theorem, which also implies that either [α, δ] = 0 or [α, [α, [α, ι]]] and [a, δ] are in two distinct summands of π_{5}(A). Thus λ_{1} = 0 and g′ = λ_{2}δ.

Next study [α, δ] ∈ π_{6}(P^{3}). Consider the pair (P^{3}, S^{2}). By the long exact homotopy sequence of the pair for the primes under consideration π_{5}(P^{3}) ≅ π_{5}(P^{3}, S^{2}). It is an elementary exercise to show that the 3-fold relative Whitehead product [ι_{2}, [ι_{2}, σ_{3}]] generates π_{5}(P^{3}, S^{2}), where σ_{3} is the relative attaching map for (P^{3}, S^{2}). By naturality it follows that [α, δ] is mapped to the 4-fold product [ι_{2}, [ι_{2}, [ι_{2}, σ_{3}]]] under the map π_{6}(P^{3}) ⟶ π_{6}(P^{3}, S^{2}). Let c : P^{3} ⟶ P^{3} ∨ S^{3} denote the cooperation map as above. Then

c_{∗} : π_{6}(P^{3}, S^{2}) ⟶ π_{6}(P^{3} ∨ S^{3}, S^{2})

satisfies c_{∗}([ι_{2}, [ι_{2}, [ι_{2}, σ_{3}]]]) = [ι_{2}, [ι_{2}, [ι_{2}, σ_{3}]]] + [ι_{2}, [ι_{2}, [ι_{2}, ι_{3}]]] ≠ 0. Hence [α, δ] ≠ 0 and so λ_{2} = 0 and g′ = 0 if [g′, α] = 0. This completes the proof of Theorem 1.5.

For each p ≥ 5, the group E(p), is in a sense the minimal counter example to the Cohen conjecture. Namely, consider any group of order less than 3p^{2}. Then either its Sylow p-subgroup is cyclic, in which case p-perfectness implies spherical resolvability [9], or its order is 2p^{2}, in which case it is not p-perfect.

Next we turn to the proof of Theorem 1.6. By the analysis of the homotopy type of X one has

X ≃ (S^{3} ∨ P^{3} ∨ S^{5} ∨ P^{6}) ∪_{h} e^{8}.

Thus the 6-skeleton X^{(6)} is a suspension and one can apply a theorem of D. Anick [1, Thm. 3.7 and Cor. 3.8] to the calculation of H_{∗}(ΩX).

Let h : S^{7} ⟶ X^{(6)} denote the attaching map for the 8-cell in X and let ĥ : S^{6} ⟶ ΩX^{(6)} denote the adjoint of h. Let h̄ ∈ H_{6}(ΩX^{(6)}) denote the image of the fundamental class ι_{6} under ĥ_{∗}. Then T = H_{∗}(ΩX^{(6)}) is a tensor algebra on a_{1}, b_{2}, b′_{2}, c_{4}, c′_{4} and d_{5} and h̄ is a primitive element there. Furthermore, by inspection of the E^{2}-term of the homology Eilenberg-Moore spectral sequence for X one observes that h̄ ≠ 0. Being a non-trivial element in a tensor algebra, h̄ generates a 2-sided ideal T h̄T which is a free T -module. Hence by [1],

H_{∗}(ΩX) ≅ T/T h̄T

as an algebra.

Finally we need to calculate the structure of H_{∗}(ΩBE(p^{r})_{p}^{∧}). Consider the fibration of H-spaces and H-maps given by

ΩX ⟶ ΩBE(p^{r})_{p}^{∧} ⟶ SU(3)_{p}^{∧}.

In order to complete the calculation we only need to observe that the generators x_{3} and x_{5} in H_{∗}(SU(3)_{p}^{∧}) are permanent cycles in the homology Serre spectral sequence for this fibration.

Indeed for the primes under consideration SU(3)_{p}^{∧} is homotopy equivalent as a space to a product of a 3-sphere and a 5-sphere. Since π_{7}S^{3} = 0 for such primes, the obvious projection X^{(6)} ⟶ S^{3} extends to X and the resulting map is degree 1 in cohomology. One uses the integral Serre spectral sequence for the fibration

SU(3)_{p}^{∧} ⟶^{i} X ⟶ BE(p^{r})_{p}^{∧}

to show that i_{∗} is degree p^{r} on the respective 3 and 5-classes. Thus we get a commutative diagram

[figure: ΩX; ΩBE(p^{r})_{p}^{∧}; SU(3); X; ΩS^{3}; S^{3}{p^{r}}; S^{3}; S^{3}; [p^{r}]]

Inspection on the corresponding homology Serre spectral sequences and naturality imply that x_{3} is a permanent cycle.

Next consider the class x_{5}. Since it is a primitive class and the class x_{3} does not support a non-trivial differential, it follows that if x_{5} does, then it must be transgressive. One now uses naturality in a diagram as above, replacing S^{3} by S^{5} and S^{3}{p^{r}} by S^{5}{p^{r}} to conclude that x_{5} is a permanent cycle. Thus the spectral sequence collapses and the result obtained.

In particular our argument shows that H_{∗}(ΩBE(p^{r})_{p}^{∧}) is a finitely generated (free) module over H_{∗}(ΩX). Thus it is finitely generated as an algebra. Further discussion of those examples will appear elsewhere.

6. Polynomial growth

Let A be a graded algebra over a field k. Define the Lie multiplication on A as usual by setting [x, y] = xy − (−1)^{|x||y|}yx. An n-fold Lie bracket [x_{1}, x_{2}, ⋯, x_{n}] is defined inductively by

[x_{1}, x_{2}, ⋯, x_{n}] = [[x_{1}, x_{2}, ⋯, x_{n−1}], x_{n}].

The algebra A is said to be Lie nilpotent of rank n if, in the associated Lie algebra, each n-fold Lie bracket vanishes. Lie nilpotent algebras enjoy many of the properties of commutative algebras. Some of those we use below.

The proof of Theorem 1.7 relies on the existence of fibrations of the form

Ω(SU(n)/G)_{p}^{∧} ⟶ ΩBG_{p}^{∧} ⟶ SU(n)_{p}^{∧}, (6.1)

for any finite group G. Since H_{∗}(ΩSU(n)) grows polynomially and since, by hypothesis of Theorem 1.7, so does H_{∗}(ΩBG_{p}^{∧}), the same holds for the homology of Ω(SU(n)/G)_{p}^{∧}. One can now use ideas of Felix, Halperin and Thomas, presented in [6, 7], to conclude Theorem 1.7. There are two ways to do so. For the first one should argue that the Hopf algebra H_{∗}(ΩBG_{p}^{∧}) has finite depth. However we have chosen to present a different proof, which is analogous to the classical proof of finite generation for the cohomology ring of a finite group.

Since (SU(n)/G)_{p}^{∧} has the homotopy type of a finite complex, the main theorem of [6] implies that H_{∗}(Ω(SU(n)/G)_{p}^{∧}) is a finitely generated Lie nilpotent algebra. Now the fibration 6.1 above is a fibration of H-spaces and H-maps. Thus its Serre spectral sequence is a spectral sequence of differential graded Hopf algebras over the finitely generated, Lie nilpotent Hopf algebra H_{∗}(Ω(SU(n)/G)_{p}^{∧}). Moreover, the base of the fibration is finite dimensional. Hence the spectral sequence collapses at a finite page. We start by briefly recalling some algebraic facts.

Lemma 6.1. Let A be a finitely generated Lie nilpotent algebra over a field κ. Then A is left and right Noetherian.

Proof. Let a_{1}, ⋯, a_{r} denote a set of algebra generators for A and suppose that A is Lie nilpotent of rank n. Consider the κ-algebra T_{n}[x_{1}, ⋯, x_{r}] given as the quotient algebra of the free associative algebra on x_{1}, ⋯ x_{r} by the set of relations given by the requirement that every n-fold Lie bracket vanishes. Then there is an obvious epimorphism of algebras

T_{n}[x_{1}, ⋯, x_{r}] ⟶ A,

given by sending x_{j} to a_{j}. A quotient algebra of a Noetherian algebra is always Noetherian. Thus it suffices to show that for each n and r the algebra T_{n}[x_{1}, ⋯, x_{r}] is Noetherian.

Consider the Lie algebra L_{n}[x_{1}, ⋯, x_{r}] over κ given by the Lie algebra quotient of the free Lie algebra on x_{1}, ⋯ x_{r} by the Lie ideal generated by all iterated Lie brackets of length at least n. It is obvious that L_{n}[x_{1}, ⋯, x_{r}] is finite dimensional. Furthermore, one observes immediately that T_{n}[x_{1}, ⋯, x_{r}] is precisely the universal enveloping algebra UL_{n}[x_{1}, ⋯, x_{r}]. But a universal enveloping algebra on a finite dimensional Lie algebra is always left and right Noetherian [11]. This completes the proof.

The following lemma is elementary.

Lemma 6.2. A module over a Noetherian algebra A is Noetherian if and only if it is finitely generated.

Lemma 6.3. Let A be a Lie nilpotent Noetherian algebra over a field k. Let D be a finitely generated differential graded A-module. Then H_{∗}(D) is a Noetherian A-module.

Proof. Since A is Noetherian and D finitely generated over A, D is Noetherian. Thus the submodule of cycles Z(D) is again Noetherian and hence so is H_{∗}(D).

We are now ready to prove Theorem 1.7. Let G be a finite p-perfect group such that H_{∗}(ΩBG_{p}^{∧}) grows polynomially. Then by [6] H_{∗}(Ω(SU(n)/G)_{p}^{∧}) is a finitely generated Lie nilpotent algebra. In the homology Serre spectral sequence for the fibration 6.1 above the E^{2} page is a finitely generated differential graded module over H_{∗}(Ω(SU(n)/G)_{p}^{∧}). Since H_{∗}(Ω(SU(n)/G)_{p}^{∧}) is Noetherian so is the E^{2} term. Thus by Lemma 6.3, the same applies inductively to each page of the spectral sequence. Since the sequence collapses in a finite page, we find that E^{∞} is a Noetherian module over H_{∗}(Ω(SU(n)/G)_{p}^{∧}).

It follows that the E^{∞} term is itself a finitely generated 𝔽_{p}- algebra. Furthermore, the E^{∞} term is a quotient of a finitely generated free module over a Lie nilpotent algebra. Hence it is a Lie nilpotent algebra. But it is a basic fact that if a filtered ring has a finitely generated associated graded then it is itself finitely generated. Thus H_{∗}(ΩBG_{p}^{∧}) is a finitely generated algebra. A similar argument implies Lie nilpotency of H_{∗}(ΩBG_{p}^{∧}).

References

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3. A. K. Bousfield and D. M. Kan; Homotopy Limits Completions and Localizations; LNM 304, (1972), Springer-Verlag.

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5. F. R. Cohen, T. J. Lada and J. P. May; The Homology of Iterated Loop Spaces ; LNM 533, (1976) Springer-Verlag.

6. Y. Felix, S. Halperin and J. C. Thomas; Hopf Algebras of Polynomial Growth; J. of Algebra 125, (1989), 408-417.

7. Y. Felix, S. Halperin and J. C. Thomas; Elliptic Spaces II; Enseign. Math. (2), 39 (1993), no. 1-2, 25-32.

8. I. James; Note on Cup Products; Proc. A.M.S., (1957) 374-383.

9. R. Levi; On Finite Groups and Homotopy Theory; to appear in the Memoirs of the AMS.

10. R. Levi; A counter example to a conjecture of Cohen; to appear in 1994 BCAT conference proceedings.

11. J. C. McConnell and J. C. Robson; Noncommutative Noetherian rings; John Wiley and Sons (1987)

12. L. Smith; Homological Algebra and the Eilenberg-Moore Spectral Sequence; AMS Translatl. 129, (1967), 58-93.

Mathematisches Institut, Universität Heidelberg, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany

E-mail address: rlvi@vogon.mathi.uni-heidelberg.de