Contemporary Mathematics Volume 00, XXXX

Some remarks concerning degree zero complete cohomology

Jonathan Cornick and Ian J. Leary

Abstract. We describe degree zero mod-p complete cohomology modulo its radical in purely group-theoretic terms, for members of a class of groups that includes all groups of finite virtual cohomological dimension. We give examples to show that our description does not apply to all discrete groups. We also give examples of discrete groups that are not ‘Quillen groups’ in the sense of H.-W. Henn.

Complete cohomology of discrete groups arose from the work of D. Benson, J. Carlson, G. Mislin and P. Vogel [BC,G,M]. It may be viewed as a generalization of Tate cohomology (which is defined only for finite groups) and Tate-Farrell cohomology (which is defined for groups of finite virtual cohomological dimension or ‘finite vcd’). For an introduction to Tate and Tate-Farrell cohomology, see [Br], and see [M] for an introduction to complete cohomology. The complete cohomology of G with coefficients in a commutative ring R, denoted Ĥ^{∗}(G; R), is a ℤ-graded anticommutative ring. In particular Ĥ^{0}(G; R) is a commutative ring. If G is a finite group, Ĥ^{0}(G; ℤ) is the quotient of ℤ by the ideal generated by |G|. In [AC], A. Adem and J. Carlson pointed out that Ĥ^{0}(G; ℤ) need not be cyclic when G is not finite, and asked for a group-theoretic interpretation of the extra cyclic summands.

Our work is motivated by this question, although our results are most easily stated for complete cohomology with coefficients in the prime field 𝔽_{p}, which we shall denote by Ĥ^{∗}(G; 𝔽_{p}) or Ĥ^{∗}(G). We give a group-theoretic description of Ĥ^{0}(G) modulo its nilradical. This description is valid for all groups G that admit a finite-dimensional contractible G-CW-complex in which all stabilizers are finite — these are the groups in P. H. Kropholler’s class H_{1}𝔉 [Kr]. We also give examples to show that our result does not hold for arbitrary discrete groups, and examples of groups of type H_{1}𝔉 for which Ĥ^{0}(G) contains elements of arbitrary nilpotency class.

The technique that we use to exhibit some of these examples can also be used to construct groups for which D. Quillen’s theorem (which describes the mod-p

\begin{matrix}c \\ ◯ \end{matrix}XXXX American Mathematical Society

1991 Mathematics Subject Classification. Primary 20J06; Secondary 57S30.

Key words and phrases. Complete cohomology, Tate-Farrell cohomology.

The second named author acknowledges partial support from the Nuffield Foundation and from EPSRC.

cohomology ring up to nilpotents in terms of the cohomology rings of elementary abelian p-subgroups) does not hold. We therefore discuss such examples, although they are not closely related to the rest of the paper. The authors thank Hans-Werner Henn for drawing their attention to the problem of exhibiting non-Quillen groups.

Many of our results and examples could be deduced fairly easily from work contained in K. S. Brown’s book [Br], except that Brown works with Tate-Farrell cohomology so that the hypothesis that G be in H_{1}𝔉 would have to be replaced by the hypothesis that G be of finite vcd. For an example to show that ‘H_{1}𝔉’ is a weaker condition than ‘finite vcd’, consider the group constructed in [Sch]. This group is finitely presented, contains exactly p elements of finite order, admits a cellular action on a contractible 2-complex with each cell stabilizer having order p, and yet does not contain any finite-index subgroups, so cannot have finite vcd.

Definition 1. For G a group, let S_{p}(G) denote the partially ordered set (or poset) of non-trivial p-subgroups of G, and let S_{p}(G)/G denote the quotient of S_{p}(G) by the conjugation action of G, or equivalently the poset of conjugacy classes of non-trivial p-subgroups of G.

Recall that a component of a poset is an equivalence class under the equivalence relation generated by the partial order, or equivalently (the vertex set of) a component of the simplicial realization of the poset.

Proposition 2. Let G be a group in H_{1}𝔉, and let J denote the nilradical of Ĥ^{0}(G). Then Ĥ^{0}(G)/J is isomorphic to a direct product of copies of 𝔽_{p} indexed by the components of S_{p}(G)/G. If there is an n-dimensional contractible G-CW-complex with finite stabilizers, then J^{n+1} = {0}.

Proof. By theorem 1.5 of [CK], there is a complete resolution of ℤ over ℤG. Let P_{∗} be such a resolution, and let Q_{∗} denote the cellular chain complex of an ndimensional contractible G-CW-complex with finite stabilizers. As in section X.4 of [Br] there is an equivariant cohomology spectral sequence arising from the double cochain complex

E_{0}^{i,j} = Hom_{G}(Q_{i} ⊗ P_{j}, 𝔽_{p}),

converging to a filtration of Ĥ^{∗}(G). In this spectral sequence, d_{0} is induced by the differential on P_{∗} and d_{1} induced by the differential on Q_{∗}. Moreover, the spectral sequence admits a product with respect to which the differentials are derivations, and from the E_{2}-page onwards the product is associative.

Using Shapiro’s lemma it may be shown that

E_{1}^{i,j} ≅ ∏_{σ} Ĥ^{j}(G_{σ}),

where the product is indexed by a set of G-orbit representatives of i-cells in X, and G_{σ} denotes the stabilizer of the cell σ. For H a finite group, Ĥ^{0}(H) is isomorphic to 𝔽_{p} if p divides the order of H, and is trivial otherwise. It follows that E_{1}^{i,0} is isomorphic to a product of copies of 𝔽_{p} indexed by those G-orbits of i-cells of X whose stabilizers have order divisible by p. Hence E_{1}^{∗,0} may be identified with the 𝔽_{p}-cochain complex on X^{(p)}/G, where X^{(p)} denotes the subcomplex of X of points stabilized by some element of order p, and E_{2}^{i,0} may be identified with H^{i}(X^{(p)}/G; 𝔽_{p}).

Now E_{2}^{0,0} is isomorphic to a product of copies of 𝔽_{p} indexed by the components of X^{(p)}/G. The higher differentials must vanish on this group, since for any r ≥ 2 and any ξ ∈ E_{r}^{0,0}, ξ^{p} = ξ, and so

d_{r}(ξ) = d_{r}(ξ^{p}) = pξ^{p−1}d_{r}(ξ) = 0.

The other groups that contribute to the filtration of Ĥ^{0}(G) in the E_{∞}-page are E_{∞}^{i,−i} for i > 0. But E_{∞}^{i,j} = 0 whenever i is greater than n, the dimension of X. Hence any ξ ∈ Ĥ^{∗}(G) that corresponds to an element of E_{∞}^{i,j} for i > 0 satisfies ξ^{n+1} = 0. Since no element of E_{∞}^{0,0} is nilpotent, it follows that for each m > 0, any ξ ∈ J^{m} corresponds to an element in E_{∞}^{i,−i} for some i ≥ m, and hence that J^{n+1} = {0} as claimed.

It remains to identify the components of X^{(p)}/G with those of S_{p}(G)/G. If H is a p-subgroup of G, P. A. Smith theory tells us that the H-fixed point set X^{H} is 𝔽_{p}- acyclic. In particular both X^{H} and its image X^{H}/G in X^{(p)}/G are connected. Now X^{(p)}/G is a union of the connected sets X^{H}/G as H ranges over the p-subgroups of G, and any two such sets X^{H}/G, X^{K}/G intersect if and only if there is a point x of X fixed by some conjugate of H and some conjugate of K. The stabilizer G_{x} of x is a finite group, and so it follows that both H and K must be conjugate to subgroups of a Sylow-p subgroup L of G_{x}, and hence that X^{L}/G ⊆ X^{H}/G ∩ X^{K}/G. Each components of X^{(p)}/G is the union of an equivalence class of X^{H}/G’s under the equivalence relation generated by ‘has non-empty intersection with’, but the above argument shows that this equivalence relation is equal to the one generated by ‘is contained in’, and equivalence classes for this relation are in 1-1 correspondence with the components of the poset S_{p}(G)/G. □

Example 3. (Discrete groups for which the analogue of Proposition 2 does not hold.) Let G be a torsion-free group whose cohomological dimension over 𝔽_{p} is infinite. If Proposition 2 applied to this group, it would follow that Ĥ^{0}(G) = {0}. But a lemma due to Kropholler [Kr] states that Ĥ^{0}(H; R) = {0} if and only if H has finite cohomological dimension over R.

For a more explicit example, let G be a free abelian group of countably infinite rank. It may be shown that for any projective ℤG-module, M, H^{∗}(G; M) = {0}. This implies that for any R, Ĥ^{∗}(G; R) ≅ H^{∗}(G; R). Hence for this G, Ĥ^{0}(G) ≅ 𝔽_{p}, although G contains no non-trivial p-subgroups.

Example 4. (Graphs of finite groups.) Take a graph of finite groups, i.e., a directed graph Γ whose vertices and edges are labelled by finite groups, together with injective homomorphisms from each edge group to its initial and terminal vertex groups. Let G be the fundamental group of this graph of finite groups, as defined in [DD]. This G acts on a tree T with finite stabilizers in such a way that there is an isomorphism from T/G to Γ. Moreover, the stabilizer of any edge or vertex of T is a subgroup of G isomorphic to the label of the corresponding cell of Γ.

Let Γ_{p} be the subgraph of Γ consisting of those edges and vertices whose labelling group has order divisible by p. It follows from Proposition 2 that the radical J of Ĥ^{0}(G) satisfies J^{2} = {0}, and that Ĥ^{0}(G)/J is isomorphic to a direct product of copies of 𝔽_{p} indexed by the components of Γ_{p}. But more can be said in this case, since the equivariant cohomology spectral sequence for a 1-dimensional G-space collapses at the E_{2}-page. For a finite group H, Ĥ^{−1}(H) is isomorphic to 𝔽_{p} if p divides |H|, and is trivial otherwise. It follows that in the spectral sequence for the action of G on T, E_{1}^{i,−1} is isomorphic to the mod-p i-cochains on the graph Γ_{p}, and hence that J ≅ E_{2}^{1,−1} is isomorphic to H^{1}(Γ_{p}; 𝔽_{p}), the first cohomology group of the space Γ_{p}. In particular, J = {0} if and only if Γ_{p} contains no cycles.

Before constructing our next family of examples, it will be helpful to quote the following version of the Kan-Thurston theorem, which is due to G. Baumslag, E. Dyer and A. Heller [BDH].

Theorem 5. For any connected CW-complex X, there is a torsion-free group G_{X} such that with any trivial coefficient system, the cohomology of the group G_{X} is isomorphic to that of the space X. Furthermore, if X is a finite simplicial complex, G_{X} may be chosen to have a finite classifying space.

We remark that J.-C. Hausmann has proved an even stronger finiteness result than that stated above: in the case when X is finite, G_{X} may be chosen to be a duality group [H].

Example 6. (Elements of high nilpotency class in Ĥ^{0}.) Suppose that G is a direct product, G = H × K, where H is a finite group, and K is a group of finite cohomological dimension. Under these circumstances, K. S. Brown shows that there is an isomorphism of rings between Ĥ^{∗}(G) and Ĥ^{∗}(H) ⊗ H^{∗}(K; 𝔽_{p}) ([Br], p.276). Let H be cyclic of order p, so that Ĥ^{−2}(H) contains a unit in the ring Ĥ^{∗}(H). For any n ≥ 1, let K be a group with a finite classifying space, whose cohomology is the same as that of n-dimensional complex projective space. (The existence of such a K is guaranteed by Theorem 5.) Thus H^{∗}(K) is isomorphic to a truncated polynomial ring 𝔽_{p}[x]/(x^{n+1}), where x has degree 2. Now G = H × K has the property that Ĥ^{0}(G) contains an element of nilpotency class n.

As remarked in the introduction, the method used to construct Example 6 can also be used to exhibit groups for which the analogue of D. Quillen’s F-isomorphism theorem [Q] does not hold. For G a discrete group, let A(G) denote the category whose objects are elementary abelian p-subgroups of G (including the trivial group), and whose morphisms are those group homomorphisms that are induced by inclusion and conjugation by elements of G. There is a map α = α(G) from H^{∗}(G) = H^{∗}(G; 𝔽_{p}) to the inverse limit over E ∈ A(G) of H^{∗}(E). Following H.-W. Henn [He], say that G is a Quillen group if the kernel of α consists of nilpotent elements, and for any element ξ of the inverse limit, there exists n such that ξ^{p^{n}} is in the image of α. (Theorem 14.1 of [Q] implies that every group of finite vcd is a Quillen group.)

Example 7. (Non-Quillen groups.) Let C be a cyclic group of order p, let X be a classifying space for C, and let G be a torsion-free group whose mod-p cohomology is isomorphic to that of X. For this group the kernel of α(G) does not consist of nilpotent elements.

For C as above, let X be an Eilenberg-Mac Lane space of type K(C, 2), and let H be a torsion-free group whose mod-p cohomology is isomorphic to that of X. Let ϕ : H^{∗}(H) → H^{∗}(X) denote an isomorphism. Let ι be a generator for H^{2}(X), and define G as a central extension with kernel C and quotient H, with extension class ϕ^{−1}(ι). One may view BG, a classifying space for G, as the total space of the principle BC-bundle pulled back from the path-loop fibration over K(C, 2). Since this map of BC-bundles induces a mod-p cohomology isomorphism between the base spaces, it follows that the induced map from BG to the total space of the path-loop fibration over K(C, 2) is a mod-p cohomology isomorphism, and hence G is 𝔽_{p}-acyclic. Since C contains all the elements of G of finite order, and is central in G, the inverse limit over A(G) may be identified with H^{∗}(C), and the image of α(G) may be identified with H^{0}(C). Thus for this choice of G, there is ξ of degree two in the inverse limit such that no power of ξ is in the image of α(G).

The groups discussed in Example 7 appear in [BDH], although the fact that they are non-Quillen groups is not mentioned there. There is also a more explicit construction of an acyclic group containing only p elements of finite order, each of which is central, due to A. J. Berrick [Be].

References

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[Ha] J.-C. Hausmann, Every finite complex has the homology of a duality group, Math. Ann. 275 (1986), 327–336.

[He] H.-W. Henn, Unstable modules over the Steenrod algebra and cohomology of groups, Proceedings of Symposia in Pure Mathematics, vol. 63, 1998, pp. 277–300.

[Kr] P. H. Kropholler, On groups of type FP_{∞}, J. Pure Appl. Algebra 90 (1993), 55–67.

[M] G. Mislin, Tate cohomology for arbitrary groups via satellites, Topology Appl. 56 (1994), 293–300.

[Q] D. Quillen, The spectrum of an equivariant cohomology ring: II, Ann. Math. 94 (1971), 573–602.

[Sch] H. R. Schneebeli, On virtual properties and group extensions, Math. Z. 159 (1978), 159–167.

Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, Ohio 43210

    E-mail address: cornick@math.ohio-state.edu

Faculty of Mathematical Studies, University of Southampton, Southampton SO17 1BJ, United Kingdom

    E-mail address: ijl@maths.soton.ac.uk