ON PERIODIC (CO)HOMOLOGY OF GROUPS

OLYMPIA TALELLI

Abstract. Here we show that a countable group G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by cup product with an element in H^{q}(G, ℤ) if and only if G has periodic homology of period q after some steps with the periodicity isomorphisms induced by cap product with an element in H^{q}(G, ℤ). In [2] J. Asadollahi, A. Hajizamani and Sh. Salarian showed that if a group G is such that every flat ℤG-module has finite projective dimension then G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by cup product with an element in H^{q}(G, ℤ) if and only if G has periodic homology of period q after some steps with the periodicity isomorphisms induced by cap product with an element in H^{q}(G, C), where C is the cotorsion envelope of the trivial ℤG-module ℤ.

1. Introduction

A group G is said to have periodic cohomology after some steps if there is a positive integer q and a nonnegative integer k such that the functors H^{i}(G, ) and H^{i+q}(G, ) are naturally equivalent for all i > k. Then it is said that G has periodic cohomology of period q after k steps.

This definition coincides with the classical one when G is finite [13], i.e. there is an element g in H^{q}(G, ℤ), such that cup product with g

∪g : Ĥ^{i}(G, ) → Ĥ^{i+q}(G, )

is an isomorphism, for all i ∈ ℤ, where Ĥ^{∗}(G, ) is the Tate cohomology of G. In this case it is said that G has periodic cohomology (i.e. k = 0) [5, 4]. When G is infinite, however, k > 0 [13].

The interplay between periodicity in cohomology after some steps for a group G and free actions of G on finite dimensional CW-complexes homotopy equivalent to a sphere goes back to R. G. Swan [12] who proved that if a finite group G has periodic cohomology then G acts freely and simplicially on a finite simplicial complex homotopy equivalent to a sphere.

In [1] Adem and Smith showed that if a group G has periodic cohomology of period q after some steps and the periodicity isomorphisms are induced by cup product with an element in H^{q}(G, ℤ) then G admits a free finite dimensional G-CW-complex homotopy equivalent to a sphere.

Adem-Smith’s theorem had been proved for the class of groups of finite virtual cohomological dimension by Connolly-Prassidis in [6] and by Prassidis in [11] and Mislin-Talelli in [10] for certain classes of groups which properly contain the class of groups of finite virtual cohomological dimension.

Note that periodicity in cohomology after some steps for a group G is equivalent to the existence of a projective resolution for G, which is periodic after some steps [8, 13]. However, it is not known whether the periodicity isomorphisms are always induced by cup product with an element in H^{i}(G, ℤ).

It was shown in [14] that if a group G has periodic cohomology of period q after some steps then the periodicity isomorphisms are induced by cup product with an element in H^{q}(G, ℤ) if and only if silpℤG < ∞, where silpℤG is the supremum of the injective lengths of the projective ℤG-modules.

In [2] J. Asadollahi, A. Hajizamani and Sh. Salarian showed that if a group G is such that every flat ℤG-module has finite projective dimension then G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by cup product with an element in H^{q}(G, ℤ) if and only if G has periodic homology of period q after some steps with the periodicity isomorphisms induced by cap product with an element in H^{q}(G, C), where C is the cotorsion envelope of the trivial ℤG-module ℤ.

Note that a group G is said to have periodic homology of period q after k-steps if the functors H_{i}(G, ) and H_{i+q}(G, ) are naturally equivalent for all i > k [2].

For the proof of the above result the authors used certain flat resolutions with nice properties whose existence follows from a theorem by L. Bican, R. El Bashir and E. Enochs in [3] which says that every module admits a flat cover.

Here we show: Theorem: Let G be a countable group. Then G has periodic cohomology of period q after some steps with the periodicity isomorphisms induced by cup product with an element in H^{q}(G, ℤ) if and only if G has periodic homology of period q after some steps with the periodicity isomorphisms induced by cap product with an element in H^{q}(G, ℤ).

For the proof of the Theorem we will use the fact (Proposition) that if a group G satisfies the homology hypotheses of the Theorem then fin.f.dimℤG < ∞, where fin.f.dimℤG is the supremum of the flat dimensions of those ℤG-modules which have finite flat dimension.

Proposition: Let G be a group with periodic homology of period q after some steps and assume that the periodicity isomorphisms are induced by cap product with an element in H^{q}(G, ℤ). Then fin.f.dimℤG < ∞.

2. Proofs

Proof of the Proposition

Let P_{∗} : ⋯ →P_{i} ⟶^{ϑ_{i}} P_{i−1}→ ⋯ →P_{0}→ℤ→0 be a projective res-

olution of G and let R_{i} = imϑ_{i}, i > 0. By hypothesis there is a non-negative integer k and an element g in H^{q}(G, ℤ) such that

∩g : H_{i+q}(G, )→H_{i}(G, )

is an isomorphism for all i > k.

So if [f] = g, f : R_{q}→ℤ

[figure: ⋯; P_{q+1}; P_{q}; P_{q−1}; ϑ_{q}; R_{q}; f; i_{q}; ⋯; P_{0}; ℤ; 0; ⋯; P_{1}; P_{0}; ℤ]

then f lifts to a chain map : P_{∗}→P_{∗} of degree −q which induces cap product with g = [f].

Let

[figure: R_{q}; P_{q−1}; ℤ; A; i_{q}; λ; f; μ]

be a push-out diagram. Then it it well known that [f] = g ∈ H^{q}(G, ℤ) = Ext_{ℤG}^{q}(ℤ, ℤ) is represented by the q-extension

0→ℤ→A→P_{q−2}→ ⋯ →P_{0}→ℤ→0.

If K = ker f, then since f induces the homology periodicity isomorphisms after k steps, it follows that the flat dimension of K is finite, i.e. f.d_{ℤG}K < ∞. As K ≃ ker μ, it follows that

f.d_{ℤG}A < ∞.

So we have the following short exact sequence of ℤG-modules

(∗) 0→ℤ →^{λ} A→Ā→0

with f.d_{ℤG}A < ∞. Since Ā ≃ R_{q−1} it follows that it is ℤ-free and so A is ℤ-free too.

Assume that f.d_{ℤG}A = n. We will show that fin.f.dimℤG ≤ n + 1. Let B be a ℤG-module which is ℤ-free with f.d_{ℤG}B = m. We will show that m ≤ n. If

0→P′_{n}→ ⋯ →P′_{0}→A→0

is a flat resolution of A then

0→P′_{n} ⊗ B→ ⋯ →P′_{0} ⊗ B→A ⊗ B→0

is an exact sequence of ℤG-modules (diagonal action) since A is ℤ-free, and actually a flat resolution of A⊗B since P′_{i} ⊗B is a flat ℤG-module as B is ℤ-free.

Hence f.d_{ℤG}A ⊗ B ≤ n. Similarly it follows that f.d_{ℤG}A ⊗ B ≤ m and f.d_{ℤG}Ā ⊗ B ≤ m, since f.d_{ℤG}B = m. Consider the short exact sequence of ℤG-modules

(∗∗) 0→B→A ⊗ B→Ā ⊗ B→0

which we obtain from (*) by tensoring with B. If f.d_{ℤG}B = m > n = f.d_{ℤG}A then it follows from the long exact Tor sequence associated to (**) that f.d_{ℤG}A ⊗ B = m, a contradiction. Hence m ≤ n.

The result now follows since if M is a ℤG-module with f.d_{ℤG}M < ∞ and 0→Λ→P →M→0 is a projective presentation of M, then Λ is ℤfree and f.d_{ℤG}Λ = f.d_{ℤG}M − 1.

Proof of the Theorem

If G has periodic cohomology of period q after k-steps then there is an exact sequence of ℤG-modules of the form

0→R_{k}→P_{k+q−1}→ ⋯ →P_{k} →^{ϑ_{k}} P_{k−1}→ ⋯ →P_{0}→ℤ→0

where R_{k} = imϑ_{k} [13].

This gives rise to a projective resolution of G which is periodic of period q after k-steps

⋯ →P_{k+1} = P_{k+q+1}→P_{k} = P_{k+q}→P_{k+q−1}→ ⋯ →P_{k} →^{ϑ_{k}} P_{k−1}→ ⋯ →P_{0}→ℤ→0.

Since the periodicity isomorphisms are induced by cup product with an element in H^{q}(G, ℤ) it follows from [14, Theorem 3.2, Corollary 3.3] that silpℤG < ∞. Hence we obtain the following diagram

[figure: ⋯; P_{k+q} = P_{k}; P_{k+q−1}; R_{k}; R_{k}; ⋯; P_{k}; P_{k−1}; ⋯; P_{q}; P_{q−1}; R_{q}; f; ⋯; ⋯; P_{0}; ℤ; 0; P_{k}; P_{k−1}; R_{k}; ⋯; P_{0}; ℤ]

If g = [f] then clearly cap product with g

∩g : H_{i+q}(G, )→H_{i}(G, )

is an isomorphism for all i > k.

Now if G has periodic homology with period q after some steps and the periodicity isomorphisms are induced by cap product with an element g in H^{q}(G, ℤ) then by the Proposition and its proof it follows that g is represented by a q-extension of the form

0→ℤ→A→P_{q−2}→ ⋯ →P_{0}→ℤ→0

with P_{i} projective ℤG-modules and f.d_{ℤG}A < ∞. Moreover it follows that fin.f.dimℤG < ∞. Now since G is countable it follows from [7, Proposition 2.8] that the finitistic dimension of ℤG is finite, findim ℤG < ∞, where the finitistic dimension of ℤG is defined as the supremum of the projective dimensions of those ℤG-modules which have finite projective dimension. It now follows from a result of Jensen [9, Proposition 6] that every flat ℤG-module has finite projective dimension. Hence pd_{ℤG}A < ∞ and thus cup product with g induces periodicity isomorphisms in cohomology after pd_{ℤG}A-steps.

References

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Olympia Talelli University of Athens Department of Mathematics Panepistimiopolis, GR-157 84, Athens Greece otalelli@math.uoa.gr