A bound on the exponent of the cohomology of BC-bundles I. J. Leary Centre de Recerca Matematica, Institut d’Estudis Catalans, Apartat 50, E-08193 Bellaterra.

We give a lower bound for the exponent of certain elements in the integral cohomology of the total spaces of principal BC-bundles for C a finite cyclic group. We are mainly interested in the case when the total space is BG for some discrete group G having a central subgroup isomorphic to C. As applications we give a proof of the theorem of A. Adem and H.-W. Henn that a p-group is elementary abelian if and only if its integral cohomology has exponent p, and we exhibit some infinite groups of finite virtual cohomological dimension whose Tate-Farrell cohomology contains torsion of order greater than the l.c.m. of the orders of their finite subgroups. Our examples include a class of groups having similar properties discovered by Adem and J. Carlson. As a third application, we examine the integral cohomology of a class of p-groups expressible as central extensions with cyclic kernel and quotient abelian of p-rank two. For each such G we determine the minimal n such that almost all (i.e. all but possibly finitely many) of the groups H^{i}(BG) have exponent dividing p^{n}. The lemma we use to give an upper bound for the exponents of almost all of the groups H^{i}(BG) applies to any p-group and may be of independent interest. Here, and throughout the paper, the coefficients for cohomology are to be the integers when not otherwise stated, and we write Z_{n} for the integers modulo n. The author gratefully acknowledges that this work was funded by the DGICYT.

Proposition 1. Let C be a cyclic group of order n, and let E be a principal BC-bundle over a connected space X, classified by ξ ∈ H^{2}(X; C) of order m. Then for any i ≥ 0, any element of H^{2i}(E) restricting to the fibre as a generator for H^{2i}(BC) has order divisible by mn.

Remark. Note that we do not claim that such elements always exist, nor do we rule out the possibility that they have infinite order. Proof. In [4] Cartan and Eilenberg computed the ring H^{∗}(BC; R) for any coefficient ring R. Recall that we have the following ring isomorphisms:

H^{∗}(BC) ≅ Z[z]/(nz), H^{∗}(BC; Z_{n}) ≅ Z_{n}[x, y]/(ny, nx, y^{2} − ex),

where e = 0 if n is odd and e = n/2 if n is even, and y has degree 1 while x and z have degree two. The natural map from integral to mod-n cohomology sends z to x, and if we let β stand for the Bockstein for the coefficient sequence

0 → Z → Z → Z_{n} → 0,

then it is easy to see that β(y) = z, and that therefore β(yx^{i}) = z^{i+1}.

Now consider the spectral sequence for the given fibration with coefficients in Z_{n}. By assumption the fundamental group of X acts trivially on the cohomology of BC, and so

E_{2}^{i,j} ≅ H^{i}(X; Z_{n}) ⊗ H^{j}(BC; Z_{n}).

Now 1⊗yx^{j} represents a generator for E_{2}^{0,2j+1} and 1⊗x^{j} represents a generator for E_{2}^{0,2j}. Comparing this spectral sequence with the spectral sequence for the path-loop fibration over an Eilenberg-MacLane space K(C, 2) it is easy to see that d_{2}(1 ⊗ y) = ξ and d_{2}(1 ⊗ x) = 0. (In fact, d_{3}(1 ⊗ x) = ξ′ ⊗ 1, where ξ′ is the image of β(ξ) under the map from H^{3}(X) to H^{3}(X; Z_{n}), and d_{4} may be described using the argument given in [8], but we do not need this here.) Now d_{2}(1 ⊗ x^{j}y) = ξ ⊗ x^{j} and d_{2}(1 ⊗ x^{j}) = 0, from which it follows that E_{3}^{0,2j} is generated by 1 ⊗ x^{j} and E_{3}^{0,2j+1} by m(1 ⊗ yx^{j}). The map from H^{∗}(E; Z_{n}) to H^{∗}(BC; Z_{n}) factors through E_{∞}^{0,∗}, which is a subgroup of E_{3}^{0,∗}, and so we see that the image of H^{2j+1}(E; Z_{n}) in H^{2j+1}(BC; Z_{n}) must be contained in the subgroup generated by myx^{j}.

Now recall that the image of the Bockstein β defined above is exactly the elements of integral cohomology of order dividing n. Let f : BC → E be the inclusion of the fibre of the above fibration. Now let χ be an element of H^{∗}(E) such that f^{∗}(χ) = z^{j+1} for some j. If χ has infinite order then there is nothing to prove. Otherwise, the order of χ must be a multiple of n (the order of z^{j+1}), say m′n, and it remains to show that m divides m′. Now m′χ has order n, so there exists χ′ ∈ H^{2j+1}(E; Z_{n}) such that β(χ′) = m′χ. However, the spectral sequence argument shows that f^{∗}(χ′) is in the subgroup of H^{2j+1}(BC; Z_{n}) generated by myx^{j} and hence βf^{∗}(χ′) is in the subgroup of H^{2j+2}(BC) generated by mz^{j+1}, but βf^{∗}(χ′) = f^{∗}β(χ′) = f^{∗}(χ) = m′z^{j+1}.

Corollary 1. Let C be a cyclic subgroup of order n of a group G. If there exists an element of H^{∗}(BG) of order n whose image in H^{∗}(BC) is a generator for H^{2i}(BC) for some i, then C is a direct factor of its centraliser in G.

Proof. This is just Proposition 1 applied to the principal BC-bundle with total space the classifying space of the centraliser of C.

Corollary 2. Let G be a discrete group expressible as a central extension with kernel C cyclic of order n. Let Q be the quotient G/C, and let the extension class of G in H^{2}(BQ; C) have order m. If G has a normal subgroup N of finite index whose intersection with C is trivial (for example, if G is finite or residually finite), then for infinitely many i, H^{2i}(BG) contains elements of order mn.

Remark. The condition that the extension class of G has order m may be rephrased as follows: If D is the smallest subgroup of C such that G/D is isomorphic to (C/D) × Q, then D has order m.

Proof. Let G′ be the quotient G/N, and let C′ be the image of C in G′. Then C′ is isomorphic to C and G′ is finite. By either Evens’ argument using the Norm map from H^{∗}(BC) to H^{∗}(BG) [5,6] or Venkov’s argument using Chern classes of a representation of G′ restricting faithfully to C′ [10], we see that for infinitely many i there exists χ′ ∈ H^{2i}(BG′) whose image in H^{2i}(BC′) is a generator. If χ is the image of χ′ in H^{∗}(BG), then χ has finite order (dividing the order of G′) and its image in H^{2i}(BC) is a generator. Hence by Proposition 1, some multiple of χ has order exactly mn.

The first example of a group whose Tate-Farrell cohomology contains elements of order greater than the l.c.m. of the orders of its finite subgroups is due to Adem [2]. The following application of Corollary 2 is more closely related to some other examples due to Adem and Carlson [3]. In particular, Corollary 3 may be compared with Theorem 3.1 of [3], which gives stronger cohomological information about a smaller class of groups.

Corollary 3. With notation and hypotheses as in Corollary 2, assume also that Q has finite cohomological dimension (or equivalently, assume that there is a finite-dimensional CW-complex BQ). Then

a) G has finite virtual cohomological dimension and hence the Tate-Farrell cohomology groups Ĥ^{i}(G) are defined,

b) C consists of all the elements of G of finite order, and

c) Ĥ^{i}(G) contains elements of order mn for infinitely many i.

Proof. The subgroup N of G has finite index and is isomorphic to a subgroup of Q, so has cohomological dimension less than or equal to that of Q. Hence G has finite vcd. The group Q is torsion-free, and so any element of G − C has infinite order because its image in Q does. If i is greater than vcdG then Ĥ^{i}(G) is isomorphic to H^{i}(BG), and so the third claim follows from Corollary 2.

The following Corollary is due to Adem [1] and Henn [7].

Corollary 4. Let G be a finite p-group. Then G is not elementary abelian if and only if H^{i}(BG) contains elements of order p^{2} for some i if and only if H^{i}(BG) contains elements of order p^{2} for infinitely many i.

Proof. If G is elementary abelian (i.e. is isomorphic to a product of cyclic groups of order p) then H^{i}(G) has exponent p for i > 0 by the Künneth theorem. Conversely, if G is not elementary abelian then G contains a central subgroup of order p which is not a direct factor, or equivalently, C of order p such that the extension class of G in H^{2}(BG/C; C) has order p. The result now follows by applying Corollary 2.

The following application of Proposition 1 is new.

Proposition 2. For positive integers α, β, γ, δ satisfying the inequalities 0 ≤ γ − δ ≤ min{α, β}, let G = G(α, β, γ, δ) be a p-group with the following presentation.

G = ⟨a, b, c | [a, c] = [b, c] = 1 = a^{p^{α}} = b^{p^{β}} = c^{p^{γ}} , [a, b] = c^{p^{δ}}⟩

Now let ϵ be max{α, β, 2γ − δ}. Then for infinitely many i, H^{i}(BG) has exponent p^{ϵ}, and at most finitely many of the groups H^{i}(BG) have higher exponent.

Remark. It is easy to see that any group having a presentation of the above form for arbitrary (α, β, γ, δ) also has a presentation of the above form in which the inequalities are satisfied: If γ is less than δ, then c^{p^{δ}} = c^{p^{γ}} = 1, and so in this case G(α, β, γ, δ) is isomorphic to G(α, β, γ, γ). On the other hand, the order of [a, b] = c^{p^{d}} is bounded by the orders of a and b given that c is central, and so the order of c is bounded by p^{α+δ} and p^{β+δ}. Thus given a presentation as above but not satisfying the second inequality we could replace γ by γ′ = min{α+δ, β+δ} and obtain another presentation of the same group. Proof. First we recall that for any G and any split surjection from G onto Q, H^{∗}(BQ) occurs as a direct summand of H^{∗}(BG). Now the above group G may be expressed as a split extension with kernel ⟨a, c⟩ and quotient ⟨b⟩ ≅ Z/p^{β}, or as a split extension with kernel ⟨b, c⟩ and quotient ⟨a⟩ ≅ Z/p^{α}. Hence we deduce that H^{2i}(BG) has elements of exponents p^{α} and p^{β} for all i > 0.

G may also be viewed as a central extension with kernel ⟨c⟩ which is isomorphic to Z/p^{γ}, and quotient isomorphic to Z/p^{α} ⊕ Z/p^{β} generated by the images of a and b. The extension class of this extension is easily seen to have order p^{γ−δ}, and so it follows from Corollary 1 that for infinitely many i, H^{2i}(BG) contains elements of order p^{2γ−δ}.

For the partial converse, note that G has subgroups ⟨a, c⟩, ⟨b, c⟩, and ⟨a, b^{p^{γ−δ}}⟩ of index p^{α}, p^{β} and p^{2γ−δ} respectively whose intersection is trivial, and then apply the following Lemma.

Lemma 1. Let G be a (finite) p-group, let H_{1}, …, H_{k} be a family of subgroups of G such that the index |G : H_{j}| of each H_{j} is less than or equal to p^{n}, and suppose that the intersection

⋂_{g∈G,1≤j≤k} H_{j}^{g}

of the conjugates of the subgroups H_{j} is trivial. Then H^{i}(BG) has exponent dividing p^{n} for all but finitely many i.

Proof. Let Σ_{m} be the symmetric group on m symbols and let G_{n} be the Sylow p-subgroup of Σ_{p^{n}}. Since the index of (Σ_{m})^{p} in Σ_{mp} divides exactly once by p an easy induction argument using the transfer shows that for all i > 0 and all n, H^{i}(BG_{n}) has exponent dividing p^{n}. If H is a subgroup of G, then the kernel of the permutation representation of G on the cosets of H is the intersection of the conjugates of H. Hence if G has subgroups H_{1}, …, H_{k} as in the statement then G occurs as a subgroup of a product of k symmetric groups on at most p^{n} symbols, and hence as a subgroup of (G_{n})^{k}. The result now follows from the observation due to Adem [1] that for any group G′ and any subgroup G, the finite generation of H^{∗}(BG) as an H^{∗}(BG′)-module implies that at most finitely many of the groups H^{i}(BG) can have higher exponent than the reduced cohomology H̃^{∗}(BG′).

Remark. The bound given by Lemma 1 for the exponent of almost all of the integral cohomology groups of a p-group is attained for many groups. For example, Proposition 2 shows that the bound is attained for the groups G(α, β, γ, δ). We were tempted to conjecture that the bound is always attained, but have recently found a group of order 128 whose index four subgroups intersect nontrivially and whose integral cohomology has exponent four [9]. Adem has conjectured that for G a finite group, if H^{i}(BG) contains elements of order p^{n} for some i, then it does so for infinitely many i [1], and Henn has asked if this is the case [7]. We do not know if this holds for the groups G(α, β, γ, δ).

References.

[1] A. Adem, Cohomological exponents of ZG-lattices, J. Pure and Appl. Alg. 58 (1989), 1–5.

[2] A. Adem, On the exponent of the cohomology of discrete groups, Bull. London Math. Soc. 21 (1989), 585–590.

[3] A. Adem and J. F. Carlson, Discrete groups with large exponents in cohomology, J. Pure and Appl. Alg. 66 (1990), 111–120.

[4] H. Cartan and S. Eilenberg, Homological Algebra, Princeton Univ. Press (1956).

[5] L. Evens, The cohomology ring of a finite group, Trans. Amer. Math. Soc. 101 (1961), 224–239.

[6] L. Evens, A generalization of the transfer map in the cohomology of groups, Trans. Amer. Math. Soc. 108 (1963), 54–65.

[7] H.-W. Henn, Classifying spaces with injective mod-p cohomology, Comment. Math. Helvetici 64 (1989), 200–206.

[8] I. J. Leary, A differential in the Lyndon-Hochschild-Serre spectral sequence, J. Pure and Appl. Alg. 88 (1993), 155–168.

[9] I. J. Leary, Integral cohomology of some wreath products, in preparation.

[10] B. B. Venkov, Cohomology algebras for some classifying spaces, Dokl. Akad. Nauk SSSR 127 (1959), 943–944 (in Russian).