Free Actions on Products of Spheres at High Dimensions

Osman Berat Okutan and Ergün Yalçın^{∗}

April 17, 2013

Abstract

A classical conjecture in transformation group theory states that if G = (ℤ/p)^{r} acts freely on a product of k spheres S^{n_{1}} × ⋯ × S^{n_{k}}, then r ≤ k. We prove this conjecture in the case where the dimensions {n_{i}} are high compared to all the differences |n_{i} − n_{j}| between the dimensions. 2010 Mathematics Subject Classification. Primary: 57S25; Secondary: 20J06.

1 Introduction

Let G be a finite group. The rank of G, denoted by rk(G), is defined as the largest integer s such that (ℤ/p)^{s} ≤ G for some prime p. It is known that G acts freely and cellularly on a finite complex homotopy equivalent to a sphere S^{n} if and only if rk(G) = 1. This follows from the results due to P.A. Smith [13] and R. Swan [14]. As a generalization of this, it has been conjectured by Benson-Carlson [3] that rk(G) = hrk(G) where hrk(G) is defined as the smallest integer k such that G acts freely and cellularly on a finite CW-complex homotopy equivalent to a product of k spheres. This conjecture is often referred to as the rank conjecture. Note that one direction of the Benson-Carlson conjecture is the following statement:

Conjecture 1.1. Let p be a prime. If G = (ℤ/p)^{r} acts freely and cellularly on a finite CW-complex X homotopy equivalent to S^{n_{1}}×⋯×S^{n_{k}}, then r ≤ k.

∗Partially supported by TÜBİTAK-TBAG/110T712.

This conjecture is a classical conjecture which has been studied intensely through 80’s and it has been proven that the conjecture is true under some additional assumptions. For example it is known that when the dimensions of the spheres are all equal, i.e., n = n_{1} = ⋯ = n_{k}, then the conjecture is true for all primes p and for all positive integers n except when p = 2 and n = 3, 7. This was proved by G. Carlsson [7] in the case where the G-action on the homology of X is trivial and the general case is due to Adem-Browder [2]. The p = 2 and n = 1 case was proven later in [15]. More recently, B. Hanke [11] proved that Conjecture 1.1 is true in the case where p ≥ 3 dim X, i.e., when the prime p is large compared to the dimension of the space. In this paper, we prove Conjecture 1.1 for the other extreme, i.e., when the dimensions of the spheres are high compared to all the differences between the dimensions.

Theorem 1.2. Suppose G = (ℤ/p)^{r} for a prime p and k, l are positive integers. Then there is an integer N that depends only on k, l and G such that if G acts freely and cellularly on a finite dimensional CW-complex X homotopy equivalent to S^{n_{1}} × ⋯ × S^{n_{k}} where n_{i} ≥ N and |n_{i} − n_{j}| ≤ l for all i, j, then r ≤ k.

The proof follows from a theorem of Browder [4] which gives a restriction on the order of groups acting freely on a finite dimensional CW-complex in terms of homology groups of the complex. We also use a method of gluing homology groups at different dimensions which we first saw in a paper by Habegger [10] and a crucial result on the exponents of cohomology groups of elementary abelian p-groups which is due to Pakianathan [12].

At the end of the paper we also prove a generalization of Theorem 1.2 to non-free actions which was suggested to us by A. Adem.

The paper is organized as follows: In Section 2, we list some well-known results about hypercohomology and in Section 3, we introduce Habegger’s theorem on gluing homology at different dimensions. In Section 4, we discuss the exponents of Tate cohomology groups and in Section 5, we prove Theorem 1.2 which is our main theorem.

2 Tate Hypercohomology

Let G be a finite group and M be a ℤG-module. The Tate cohomology of G with coefficients in M is defined as follows

Ĥ^{i}(G, M) := H^{i}(Hom_{G}(F_{∗}, M))

for all i ∈ ℤ, where F_{∗} is a complete ℤG-resolution of ℤ (see [5, p. 134]). We can generalize this and define Tate hypercohomology of G with coefficients in a chain complex C_{∗} of ℤG-modules. To do this, we need to extend the contravariant functor Hom_{G}(−, M) to ℋ om_{G}(−, C_{∗}). We will define it as in Brown (see [5, p. 5]), but instead of defining it as a chain complex, we consider it as a cochain complex.

Suppose C_{∗} and D_{∗} are chain complexes over ℤG with differentials ∂^{C} and ∂^{D}, respectively. For all n ∈ ℤ, let ℋ om_{G}(C_{∗}, D_{∗})^{n} denote the set of graded G-module homomorphisms of degree −n and define the boundary map δ^{n} by δ^{n}(f) = f∂^{C} − (−1)^{n}∂^{D}f. Note that ℋ om_{G}(−, C_{∗}) (resp. ℋ om_{G}(C_{∗}, −)) becomes a covariant (resp. contravariant) functor from the category of chain complexes of ℤG-modules to the category of cochain complexes of abelian groups. Also, if C_{∗} is a chain complex concentrated at 0 with C_{0} = M, then ℋ om_{G}(−, C_{∗}) is naturally equivalent to the functor Hom_{G}(−, M).

Now, we define the Tate hypercohomology of a finite group G with coefficients in C_{∗} as follows:

Ĥ^{i}(G, C_{∗}) := H^{i}(ℋ om_{G}(F_{∗}, C_{∗}))

for all i ∈ ℤ, where F_{∗} is a complete ℤG-resolution of ℤ. We immediately have Ĥ^{i}(G, ΣC_{∗}) ≅ Ĥ^{i+1}(G, C_{∗}), where (ΣC_{∗})_{i} = C_{i−1} for all i. Therefore, if C_{∗} is a chain complex concentrated at n, then Ĥ^{i}(G, C_{∗}) ≅ Ĥ^{i+n}(G, C_{n}). Also note that given a short exact sequence of chain complexes

0 → C_{∗} → D_{∗} → E_{∗} → 0

of ℤG-modules, there is a long exact sequence of the following form

⋯ → Ĥ^{i}(G, C_{∗}) → Ĥ^{i}(G, D_{∗}) → Ĥ^{i}(G, E_{∗}) → Ĥ^{i+1}(G, C_{∗}) → · · · .

An important property of ℋ om functor is that if P_{∗} is a chain complex of projective ℤG-modules and f_{∗} : C_{∗} → D_{∗} a weak equivalence of nonnegative chain complexes of ℤG-modules, then f_{∗} : ℋ om_{G}(P_{∗}, C_{∗}) → ℋ om_{G}(P_{∗}, D_{∗}) is also a weak equivalence (see [5, p. 29]). Actually, Brown proves this result by assuming P_{∗} is nonnegative and C_{∗} and D_{∗} are arbitrary, but the same proof remains true if we assume P_{∗} is arbitrary and C_{∗} and D_{∗} are nonnegative. Using this, we obtain the following proposition:

Proposition 2.1. If C_{∗} is a nonnegative chain complex of ℤG-modules with homology concentrated at dimension n and H_{n}(C_{∗}) = M, then Ĥ^{i}(G, C_{∗}) ≅ Ĥ^{i+n}(G, M).

Proof. Let Z_{n} denote the group of n-cycles in C_{∗}. We have the following weak equivalences:

D_{∗} : ⋯ ⟶ C_{n+1} ⟶ Z_{n} ⟶ 0 ⟶ ⋯
id↓  ↓  ↓
C_{∗} : ⋯ ⟶ C_{n+1} ⟶ C_{n} ⟶ C_{n−1} ⟶ ⋯

and D_{∗} : ⋯ ⟶ C_{n+1} ⟶ Z_{n} ⟶ 0 ⟶ ⋯

↓  ↓  ↓
E_{∗} : ⋯ ⟶ 0 ⟶ M ⟶ 0 ⟶ · · · .

Therefore, Ĥ^{i}(G, C_{∗}) ≅ Ĥ^{i}(G, D_{∗}) ≅ Ĥ^{i}(G, E_{∗}) ≅ Ĥ^{i+n}(G, M).

An exact sequence K ⟶^{f} L ⟶^{g} M of ℤG-modules is called admissible if the inclusion map im(g) ↪ M is ℤ-split (see [5, p. 129]). A ℤG-module M is called relatively injective if Hom_{G}(−, M) takes an admissible exact sequence to an exact sequence of abelian groups. Projective ℤG-modules are relatively injective (see [5, p. 130]). Since a complete ℤG-resolution F_{∗} of ℤ is an exact sequence of free ℤG-modules, the sequence F_{i+1} → F_{i} → F_{i−1} is admissible for all i. Hence if P is a projective ℤG-module, then the Tate cohomology group Ĥ^{i}(G, P) = 0 for all i. This result generalizes to hypercohomology.

Proposition 2.2. If P_{∗} is a chain complex of projective ℤG-modules which has finite length, then Ĥ^{i}(G, P_{∗}) = 0 for all i.

Proof. Recall that we say a chain complex C_{∗} has finite length if there are integers n and m such that C_{i} = 0 for all i > n and i < m. By shifting P_{∗} if necessary, we can assume that P_{∗} is a finite dimensional nonnegative chain complex and prove the proposition by an easy induction on the dimension of P_{∗}.

We say that two chain complexes C_{∗} and D_{∗} are freely equivalent if there is a sequence of chain complexes C_{∗} = E_{∗}^{0}, …, E_{∗}^{n} = D_{∗} such that either E_{∗}^{i} is an extension of E_{∗}^{i−1} or E_{∗}^{i−1} is an extension of E_{∗}^{i} by a finite length chain complex of free modules. Note that we say a chain complex D_{∗} is an extension of C_{∗} by a finite length chain complex of free modules if there is short exact sequence of chain complexes either of the form 0 → C_{∗} → D_{∗} → F_{∗} → 0 or of the form 0 → F_{∗} → D_{∗} → C_{∗} → 0, where F_{∗} is a finite length chain complex of free modules. As a corollary of Proposition 2.2, we have:

Corollary 2.3. If two chain complexes C_{∗} and D_{∗} are freely equivalent, then Ĥ^{i}(G, C_{∗}) ≅ Ĥ^{i}(G, D_{∗}) for all i.

Before we conclude this section, we would like to note that there is a hypercohomology spectral sequence which converges to the Tate hypercohomology Ĥ^{∗}(G, C_{∗}) for a given chain complex C_{∗} of ℤG-modules. One way to obtain this spectral sequence is to consider the double complex D^{p,q} = Hom_{G}(F_{p}, C_{−q}) where the vertical and horizontal differentials are given by δ_{0} = Hom(−, ∂) and δ_{1} = Hom(∂, −). Note that the total complex TotD^{∗,∗} with

Tot^{n}D^{∗,∗} = ⨁_{p+q=n} D^{p,q}

and δ^{n} = δ_{0} − (−1)^{n}δ_{1} is a cochain complex homotopy equivalent to the cochain complex ℋ om_{G}(F_{∗}, C_{∗}). Filtering this double complex with respect to the index p and then with respect to the index q, we obtain two spectral sequences

IE_{2}^{p,q} = Ĥ^{p}(G, H_{−q}(C_{∗})) ⇒ Ĥ^{p+q}(G, C_{∗})
IIE_{1}^{p,q} = Ĥ^{q}(G, C_{−p}) ⇒ Ĥ^{p+q}(G, C_{∗}).

Note that using these two spectral sequences it is possible to give alternative proofs for Proposition 2.1 and 2.2.

3 Habegger’s Theorem

In [10, p. 433-434], Habegger uses a technique to “glue” homology groups of a chain complex at different dimensions. This technique will be crucial in the proof of Theorem 1.2, so we give a proof for it here. Before we state Habegger’s theorem, we recall the definition of syzygies of modules.

For every positive integer n, the n-th syzygy of a ℤG-module M is defined as the kernel of ∂_{n−1} in a partial resolution of the form

P_{n−1} ⟶^{∂_{n−1}} ⋯ → P_{1} ⟶^{∂_{1}} P_{0} → M → 0

where P_{0}, …, P_{n−1} are projective ℤG-modules. We denote the n-th syzygy of M by Ω^{n}M and by convention we take Ω^{0}M = M.

The n-th syzygy of a module M is well-defined only up to stable equivalence. Recall that two ℤG-modules M and N are called stably equivalent if there are projective ℤG-modules P and Q such that M ⊕ P ≅ N ⊕ Q. Welldefinedness of syzygies up to stable equivalence follows from a generalization of Schanuel’s lemma (see [5, p. 193]). Since for any two stably equivalent modules M and N, we have Ĥ^{i}(G, M) ≅ Ĥ^{i}(G, N) for all i, we will ignore the fact that syzygies are well-defined only up to stable equivalence and treat Ω^{n}M as a unique module depending only on M and n. Alternatively, one can fix a resolution for every ℤG-module M and define Ω^{n}M as the kernel of ∂_{n−1} in this unique resolution.

Theorem 3.1 (Habegger [10]). Let C_{∗} be a chain complex of ℤG-modules and n, m are integers such that m < n. If H_{k}(C_{∗}) = 0 for all k with m < k < n, then C_{∗} is freely equivalent to a chain complex D_{∗} such that (i) H_{i}(D_{∗}) = H_{i}(C_{∗}) for every i ≠ n, m; (ii) H_{m}(D_{∗}) = 0, and; (iii) there is an exact sequence of ℤG-modules

0 → H_{n}(C_{∗}) → H_{n}(D_{∗}) → Ω^{n−m}H_{m}(C_{∗}) → 0.

Proof. Let F_{n−1} → ⋯ → F_{m} → H_{m}(C_{∗}) → 0 be an exact sequence where F_{i}’s are free ℤG-modules. Consider the following diagram

⋯ ⟶ 0 ⟶ F_{n−1} ⟶ ⋯ ⟶ F_{m} ⟶ H_{m}(C_{∗}) ⟶ 0 ⟶ ⋯
id↓  ↓
⋯ ⟶ C_{n} ⟶ C_{n−1} ⟶ ⋯ ⟶ Z_{m} ⟶ H_{m}(C_{∗}) ⟶ 0 ⟶ ⋯

where Z_{m} denotes the group of m-cycles in C_{∗}. Since all F_{i}’s are projective and the bottom row has no homology below dimension n, the identity map extends to a chain map between rows. Notice that this chain map gives a chain map f_{∗} : F_{∗} → C_{∗} as follows

⋯ ⟶ 0 ⟶ F_{n−1} ⟶ ⋯ ⟶ F_{m} ⟶ 0 ⟶ ⋯
↓  ^{f_{n−1}}↓  ^{f_{m}}↓  ↓
⋯ ⟶ C_{n} ⟶ C_{n−1} ⟶ ⋯ ⟶ C_{m} ⟶ C_{m−1} ⟶ · · · .

where the maps f_{i} : F_{i} → C_{i} for i > m are the same as the maps in the first diagram above. The map f_{m} : F_{m} → C_{m} is defined as the composition

F_{m} ⟶^{f′_{m}} Z_{m} ↪ C_{m}

where f′_{m} : F_{m} → Z_{m} is the map defined as the lifting of the identity map in the first diagram.

Now, let D_{∗} be the mapping cone of f_{∗}. We have the following short exact sequence of the form

0 → C_{∗} → D_{∗} → ΣF_{∗} → 0,

so C_{∗} is freely equivalent to D_{∗}. The corresponding long exact sequence of homology groups is

⋯ ⟶ H_{i}(F_{∗}) ⟶^{f_{∗}} H_{i}(C_{∗}) ⟶ H_{i}(D_{∗}) ⟶ H_{i−1}(F_{∗}) ⟶ · · · .

Assume first that n > m + 1. Then F_{∗} has at least two terms and its homology is nonzero only at two dimensions n − 1 and m. So, H_{i}(C_{∗}) ≅ H_{i}(D_{∗}) for all i such that i ≠ m, m + 1, n − 1, n. At dimension m, the map f_{∗} : H_{m}(F_{∗}) → H_{m}(C_{∗}) is an isomorphism, so we get H_{m}(D_{∗}) = H_{m+1}(D_{∗}) = 0. At dimension n − 1, we have H_{n−1}(C_{∗}) = 0, so we get H_{n−1}(D_{∗}) = 0. We also have a short exact sequence of the form

0 ⟶ H_{n}(C_{∗}) ⟶ H_{n}(D_{∗}) ⟶ H_{n−1}(F_{∗}) ⟶ 0.

Since H_{n−1}(F_{∗}) ≅ Ω^{n−m}(H_{m}(C_{∗})), this gives the desired result.

If n = m + 1, then F_{∗} has a single term F_{m}, so we have a sequence of the form

0 ⟶ H_{n}(C_{∗}) ⟶ H_{n}(D_{∗}) ⟶ F_{m} ⟶^{f_{∗}} H_{m}(C_{∗}) ⟶ H_{m}(D_{∗}) ⟶ 0.

Since f_{∗} is surjective by construction, we conclude that H_{m}(D_{∗}) = 0 and there is a short exact sequence of the form

0 ⟶ H_{n}(C_{∗}) ⟶ H_{n}(D_{∗}) ⟶ Ω^{1}(H_{m}(C_{∗})) ⟶ 0

as desired.

4 Exponents of Tate Cohomology Groups

To prove the main theorem, we need some results about the exponents of Tate cohomology groups. We first recall some definitions. The exponent of a finite abelian group A is defined as the smallest positive integer n such that na = 0 for all a ∈ A. We denote the exponent of A by expA. Note that if A → B → C is an exact sequence of finite abelian groups, then expB divides expA · expC. In this situation we sometimes write expB/expA divides expC to refer to the same fact even though expB/expA may not be an integer in general.

The first result we prove is a proposition on the exponent of Tate cohomology group with coefficients in a filtered module. First let us explain the terminology that we will be using throughout the paper. Let M be a ℤG-module and A_{1}, A_{2}, …, A_{n} be a sequence of ℤG-modules. If M has a filtration 0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = M such that M_{j}/M_{j−1} ≅ A_{j} for all j, then we say M has a filtration with sections A_{1} − A_{2} − ⋯ − A_{n}.

Proposition 4.1. Let M be a ℤG-module which has a filtration with sections A_{1} − A_{2} − ⋯ − A_{n}. Then, expĤ^{i}(G, M) divides ∏_{j=1}^{n} expĤ^{i}(G, A_{j}).

Proof. Let 0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{n} = M be the filtration of M with the sections as above. Then for every j, we have an exact sequence of ℤGmodules

0 → M_{j−1} → M_{j} → A_{j} → 0,

which gives a long exact Tate cohomology sequence of the following form

⋯ → Ĥ^{i}(G, M_{j−1}) → Ĥ^{i}(G, M_{j}) → Ĥ^{i}(G, A_{j}) → · · · .

From this we observe that expĤ^{i}(G, M_{j})/expĤ^{i}(G, M_{j−1}) divides the exponent of Ĥ^{i}(G, A_{j}). Multiplying these relations through all j = 1, …, n, we get expĤ^{i}(G, M) divides ∏_{j=1}^{n} expĤ^{i}(G, A_{j}).

In [4], Browder proves a theorem which gives an upper bound on the order of a finite group G in terms of the exponents of cohomology groups with coefficients in homology groups of a CW-complex on which G acts freely. Since we use this theorem in the proof of our main theorem, we state it below and give a proof for it. The proof we give here is slightly different than the original proof. It uses Theorem 3.1 and Proposition 4.1.

Theorem 4.2 (Browder [4]). Let C_{∗} be a nonnegative, free, connected chain complex of dimension n. Then |G| divides ∏_{j=1}^{n} expH^{j+1}(G, H_{j}(C_{∗})).

Proof. Let us take C_{∗}^{(0)} = C_{∗} and for j = 1 to n, define C_{∗}^{(j)} to be the chain complex obtained by C_{∗}^{(j−1)} by applying the method in Theorem 3.1 for the dimensions n−j and n. Since C_{∗} is a finite dimensional chain complex of free ℤG-modules, by Proposition 2.2, Ĥ^{i}(G, C_{∗}) = 0 for all i. Hence by Corollary 2.3 and Theorem 3.1, we have Ĥ^{i}(G, C_{∗}^{(j)}) = 0 for all i, j. Notice that C_{∗}^{(n)} is a chain complex with homology concentrated at n. Let us denote the homology of C_{∗}^{(n)} at n by M. Hence, by Proposition 2.1, we have Ĥ^{i}(G, M) = 0 for all i. By Theorem 3.1, M has a filtration

0 ⊆ H_{n}(C_{∗}^{(0)}) ⊆ ⋯ ⊆ H_{n}(C_{∗}^{(n−1)}) ⊆ H_{n}(C_{∗}^{(n)}) = M

with sections H_{n}(C_{∗}) − Ω^{1}H_{n−1}(C_{∗}) − ⋯ − Ω^{n−1}H_{1}(C_{∗}) − Ω^{n}H_{0}(C_{∗}). If we let M′ := H_{n}(C_{∗}^{(n−1)}), then M′ has a filtration with sections H_{n}(C_{∗}) − Ω^{1}H_{n−1}(C_{∗}) − ⋯ − Ω^{n−1}H_{1}(C_{∗}) and there is a short exact sequence of the form

0 → M′ ⟶ M ⟶^{π} Ω^{n}H_{0}(C_{∗}) → 0.

Note that H_{0}(C_{∗}) ≅ ℤ, so we obtain an exact sequence of the form

⋯ → Ĥ^{n}(G, M) ⟶^{π_{∗}} Ĥ^{n}(G, Ω^{n}ℤ) → Ĥ^{n+1}(G, M′) → Ĥ^{n+1}(G, M) → · · · .

Since Ĥ^{i}(G, M) = 0 for all i, we obtain Ĥ^{n+1}(G, M′) ≅ Ĥ^{n}(G, Ω^{n}ℤ) ≅ Ĥ^{0}(G, ℤ) ≅ ℤ/|G|. Hence by Proposition 4.1, we get |G| = expĤ^{n+1}(G, M′) divides the product

∏_{j=1}^{n} expĤ^{n+1}(G, Ω^{n−j}H_{j}(C_{∗})).

Since Ĥ^{n+1}(G, Ω^{n−j}H_{j}(C_{∗})) ≅ H^{j+1}(G, H_{j}(C_{∗})), this gives the desired result.

As a corollary of Theorem 4.2, Browder gives a proof for a theorem of G. Carlsson [7] which says that if G = (ℤ/p)^{r} acts freely on a finite dimensional CW-complex X ≃ (S^{n})^{k} with trivial action on homology, then r ≤ k. The main observation is that when G = (ℤ/p)^{r} and M is a trivial ℤG-module, the exponent of H^{i}(G, M) divides p for all i ≥ 1. This follows easily by induction on r using properties of the transfer map in group cohomology. So, from the relation given in Theorem 4.2, one obtains that if G acts freely on a finite dimensional CW-complex X ≃ (S^{n})^{k} with trivial action on homology, then |G| = p^{r} divides p^{k}, which gives r ≤ k.

Note that the assumption that G acts trivially on the homology of X is crucial in the above argument since for an arbitrary ℤG-module, the exponent of H^{i}(G, M) can be as large as the order of |G|. In fact, if we take M = Ω^{i}(ℤ) for some positive integer i, then we have H^{i}(G, M) ≅ ℤ/|G|, so the exponent of H^{i}(G, M) is equal to |G| in this case. Taking the direct sum of all such modules over all i, one can obtain a ℤG-module M such that the exponent of H^{i}(G, M) is equal to |G| for every i ≥ 0. The following theorem says that when M is finitely generated this situation cannot happen and that the exponent of H^{i}(G, M) eventually becomes small at high dimensions.

Theorem 4.3 (Pakianathan [12]). Let G = (ℤ/p)^{r} and M be a finitely generated ℤG-module. Then, there is an integer N such that the exponent of H^{i}(G, M) divides p for all i ≥ N.

Proof. By Theorem 7.4.1 in [9, p. 87], H^{∗}(G, M) is a finitely generated module over the ring H^{∗}(G, ℤ). Let m_{1}, ..., m_{k} be homogeneous elements generating H^{∗}(G, M) as an H^{∗}(G, ℤ)-module and let N = 1 + max_{j}{deg m_{j}}. If x ∈ H^{i}(G, M) such that i ≥ N, then we can write x = Σ_{j=1}^{k}α_{j}m_{j} for some homogeneous elements α_{j} in H^{∗}(G, ℤ) with deg α_{j} ≥ 1 for all j. Since expH^{i}(G, ℤ) divides p for all i ≥ 1, we have pα_{j} = 0 for all j. Hence we obtain px = Σ_{j=1}^{k}pα_{j}m_{j} = 0 as desired.

5 Proof of the Main Theorem

Let G = (ℤ/p)^{r} and k, l be positive integers. We will show that there is an integer N such that if G acts freely and cellularly on a CW-complex X homotopy equivalent to S^{n_{1}} × ⋯ × S^{n_{k}} where |n_{i} − n_{j}| ≤ l and n_{i} ≥ N for all i, j, then r ≤ k.

Suppose that G acts freely and cellularly on some CW-complex X homotopy equivalent to S^{n_{1}} × ⋯ × S^{n_{k}} where |n_{i} − n_{j}| ≤ l for all i, j. Let n = max{n_{i} : i = 1, …, k} and let a_{i} = n − n_{i} for all i. Consider the cellular chain complex C_{∗}(X) of the CW-complex X. The complex C_{∗}(X) is a nonnegative, connected, and finite-dimensional chain complex of free ℤGmodules and has nonzero homology only at the following dimensions other than dimension zero:

(1)  n − a_{1}, n − a_{2}, …, n − a_{k}
(2) 2n − a_{1} − a_{2}, 2n − a_{1} − a_{3}, …, 2n − a_{k−1} − a_{k}
⋮
(j) jn − (a_{1} + ⋯ + a_{j}), …, jn − (a_{k−j+1} + ⋯ + a_{k})
⋮
(k)  kn − (a_{1} + a_{2} + ⋯ + a_{k}).

If n > lk, then we have n > a_{1} + ⋯ + a_{k} which implies that for all j, the dimensions listed on the j-th row are strictly larger than the dimensions listed on the (j −1)-st row. Since this fact is crucial for our argument, we will assume that the integer N in the statement of the theorem satisfies N > lk to guarantee that this condition holds.

Now we can apply Habegger’s argument given in Theorem 3.1 to glue all the homology groups at the dimensions listed on the j-th row above to the homology at dimension jn for all j = 1, …, k. The resulting complex D_{∗} is a connected, finite-dimensional chain complex of free ℤG-modules which has homology only at dimensions 0, n, 2n, …, kn. Let M_{j} := H_{jn}(D_{∗}) for all j = 1, …, k. Note that by construction M_{j} is a finitely generated ℤGmodule for all j since syzygies of finitely generated ℤG-modules are finitely generated when G is a finite group.

Now we can apply Theorem 4.3 to find an integer N_{j} for each j such that if i ≥ N_{j}, then expH^{i}(G, M_{j}) divides p. Suppose that for a fixed G = (ℤ/p)^{r}, k, and l, there are only finitely many possibilities for ℤG-modules M_{j}’s up to stable equivalence. Then by taking the maximum of N_{j}’s over all possible M_{j}’s, we can find an integer N_{j}^{max} for each j such that if i ≥ N_{j}^{max}, then expH^{i}(G, M_{j}) divides p for all possible M_{j}’s that may occur. Then we can take N = max_{j} N_{j}^{max} and complete the proof in the following way. By Theorem 4.2, we have |G| = p^{r} divides

∏_{j=1}^{k} H^{jn+1}(G, H_{jn}(D_{∗})) = ∏_{j=1}^{k} H^{jn+1}(G, M_{j}).

So, if n ≥ N, then p^{r} divides p^{k} which gives r ≤ k as desired.

Hence to complete the proof, it only remains to show that for fixed G = (ℤ/p)^{r}, k, and l, there are only finitely many possibilities for ℤG-modules M_{j}’s up to stable equivalence. To show this, first note that for a fixed l, there are finitely many k-tuples (a_{1}, ..., a_{k}) with the property that 0 ≤ a_{i} ≤ l for all i. So we can assume that we have a fixed k-tuple (a_{1}, …, a_{k}). Let us also fix an integer j and show there are only finitely many possibilities for M_{j} = H_{jn}(D_{∗}).

Let s_{1} < ⋯ < s_{m} be a sequence of integers such that {jn−s_{1}, …, jn−s_{m}} is the set of all distinct dimensions on the j-th row of the above diagram. Note that the complex D_{∗} is constructed with the repeated usage of Theorem 3.1, so the module M_{j} = H_{jn}(D_{∗}) has a filtration

0 = K_{0} ⊆ K_{1} ⊆ ⋯ ⊆ K_{m} = M_{j}

such that K_{i}/K_{i−1} ≅ Ω^{s_{i}}(A_{i}) where A_{i} = H_{jn−s_{i}}(X). For all i, the module A_{i} is a ℤ-free ℤG-module with ℤ-rank less than or equal to \binom{k}{j}, so by Jordan-Zassenhaus theorem (see Corollary (79.12) in [8, p. 563]), there are only finitely many possibilities for A_{i}’s up to isomorphism.

We will inductively show that there exist only finitely many possibilities for K_{i}’s up to stable equivalence. For i = 1, we have K_{1} = Ω^{s_{1}}(A_{1}) so this follows from the fact that there are only finitely many possibilities for A_{1} and that syzygies are well-defined up to stable equivalence. For i > 1, consider the following short exact sequence:

0 ⟶ K_{i−1} ⟶ K_{i} ⟶ Ω^{s_{i}}A_{i} ⟶ 0.

By induction we know that there are only a finite number of possibilities for K_{i−1}’s up to stable equivalence. By a similar argument as above, the same is true for Ω^{s_{i}}(A_{i}). The extensions like the ones above are classified by the ext-group Ext_{ℤG}^{1}(Ω^{s_{i}}(A_{i}), K_{i−1}) and since both modules are ℤ-free, these ext-groups are well-defined up to stable equivalence. So, it remains to show that

Ext_{ℤG}^{1}(Ω^{s_{i}}(A_{i}), K_{i−1}) = Ext_{ℤG}^{s_{i}+1}(A_{i}, K_{i−1})

is a finite group. Note that since both A_{i} and K_{i−1} are finitely generated, Ext_{ℤG}^{s_{i}+1}(A_{i}, K_{i−1}) is a finitely generated abelian group. Moreover, since A_{i} is ℤ-free, it has an exponent divisible by |G|. So, Ext_{ℤG}^{s_{i}+1}(A_{i}, K_{i−1}) is a finite group. This completes the proof of Theorem 1.2.

We conclude this section with a generalization of Theorem 1.2 to non-free actions. The exact statement is as follows.

Theorem 5.1. Let G = (ℤ/p)^{r} and k, l be positive integers. Then there exists an integer N (depending on k, l and the group G) such that if G acts cellularly on a finite dimensional CW-complex X homotopy equivalent to S^{n_{1}} × ⋯ × S^{n_{k}} where n_{i} ≥ N and |n_{i} − n_{j}| ≤ l for all i, j, then r − s ≤ k where s is the largest integer such that |G_{x}| = p^{s} for some x ∈ X.

Proof. Let C_{∗} := C_{∗}(X) denote the cellular chain complex of X and let ε : C_{∗} → ℤ be the map induced by the constant map X → pt. The arguments in the proof of Theorem 1.2 can be repeated to prove that there is an integer N such that if n_{i} ≥ N and |n_{i} − n_{j}| ≤ l for all i, j, then

p^{k}Ĥ^{0}(G, ℤ) ⊆ im{ε_{∗} : Ĥ^{0}(G, C_{∗}) → Ĥ^{0}(G, ℤ)}.

This can be seen by a spectral sequence argument or by the filtration argument given in the proof of Theorem 4.2. To see it using the filtration argument, observe that the map ϵ_{∗} can be written as a composition

ϵ_{∗} : Ĥ^{0}(G, C_{∗}) ≅ Ĥ^{0}(G, C_{∗}^{(n)}) ≅ H^{n}(G, M) ⟶^{π_{∗}} Ĥ^{n}(G, Ω^{n}ℤ) ≅ Ĥ^{0}(G, ℤ)

where the module M and the map π_{∗} are as given in the proof of Theorem 4.2. Repeating the arguments in the proof of Theorem 1.2, we can show that there is an integer N such that if n_{i} ≥ N and |n_{i} − n_{j}| ≤ l for all i, j, then expĤ^{n+1}(G, M′) divides p^{k} where M′ is as in the proof of Theorem 4.2. Using the long exact sequence given in the proof of Theorem 4.2, we can conclude that the inclusion above holds.

The inclusion given above implies that |G| = p^{r} divides p^{k} ·expĤ^{0}(G, C_{∗}). Hence the proof will be complete if we can show that expĤ^{0}(G, C_{∗}) divides p^{s} where s is the largest integer such that |G_{x}| = p^{s} for some x ∈ X. However this is already known to be true as proven by A. Adem [1, Theorem 3.1 and 3.2]. So the proof is complete.

Acknowledgements: We thank Jonathan Pakianathan and Alejandro Adem for many helpful conversations on the paper and the referee for a careful reading of the paper and for his/her corrections and helpful comments.

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Department of Mathematics Bilkent University, Ankara, 06800, Turkey. E-mail addresses: okutan@fen.bilkent.edu.tr, yalcine@fen.bilkent.edu.tr