GROUP HOMOMORPHISMS INDUCING ISOMORPHISMS IN COHOMOLOGY

DAVE BENSON AND LEONARD EVENS

Introduction

In 1961, one of us [3] announced the following theorem (without proof).

Theorem 1. Let ϕ : G′ → G be a homomorphism of finite groups. If ϕ^{∗} : H^{∗}(G, ℤ) → H^{∗}(G′, ℤ) is an isomorphism, then ϕ is an isomorphism.

In 1977 Jackowski [5] independently arrived at the same result with a similar proof. These proofs depended on a topological construction. Using a tool introduced by Benson and Carlson in [1], we are now in a position to provide a proof entirely within the context of cohomology of groups. In fact, we prove the following result.

Theorem 2. Let G be a finite group, H a subgroup, and let M be a finitely generated torsion free G-module with M^{G} ≠ {0}. If res_{G→H} : H^{∗}(G, M) → H^{∗}(H, M) is an isomorphism, then H = G.

Theorem 1 follows from Theorem 2 as follows. Take M = ℤ. Using

G′ → H = Im ϕ < G

we may decompose ϕ^{∗} into a restriction followed by an inflation. It follows that the inflation is surjective. However, in that case the restriction from G′ to the kernel of G′ → H would be trivial in all positive degrees. The latter is impossible by [4] (and by the way the proof there uses only constructions from cohomology of groups). Hence, we are reduced to the case ϕ is a monomorphism.

The proof of Theorem 2

The heart of the proof is the following result.

Lemma 3. Let G be a finite group. There exists a finite chain complex

0 → X_{q} → X_{q−1} → ⋯ → X_{1} → X_{0} → 0

on which ℤG acts with the following properties:

(a) Each X_{i} is a finitely generated projective ℤG-module.

(b) H_{i}(X) is a trivial G-module for each i.

(c) H_{q}(X) = ℤ.

A proof based on hypercohomology will appear in [2]. A proof based on varieties is implicit in [1]. Indeed, the construction of the corresponding complex over a field K is described explicitly in section 6 of [1] as a variation on the construction of a multiply periodic resolution. Applying the methods in section 4 of that paper to that variation leads to the construction of the desired complex over ℤG. As noted in section 4 of [1], one may even arrange for the complex to be ℤG-free.

It should be noted that the complex of the lemma is easy to construct topologically. Choose a faithful unitary representation of G and thereby identify G with a finite subgroup of a unitary group U_{r}. Since G then acts freely on U_{r}, it is easy to construct a chain complex X = C_{∗}(U_{r}) such that C_{i}(U_{r}) is a finitely generated free ℤG module for each i. Since U_{r} is connected, the action of each element of G on U_{r} may be continuously deformed to the identity, so each element acts trivially on H_{∗}(U_{r}). (c) follows from the fact that U_{r} is a manifold.

To prove the theorem, choose a projective ℤG-resolution of ℤ

⋯ → P_{r} → P_{r−1} → ⋯ → P_{1} → P_{0} → ℤ → 0

and consider the double complex

Hom_{ℤG}(P ⊗ X, M) ≅ Hom_{ℤG}(P, Hom(X, M)).

Write

H^{s}(X, M) = H^{s}(Hom(X, M)),
H_{G}^{s}(X, M) = H^{s}(Hom_{ℤG}(X, M)).

There are two spectral sequences associated with this double complex. One has E_{2} term

H^{r}H^{s}(Hom_{ℤG}(P, Hom(X, M))) = H^{r}(G, H^{s}(X, M)),

and the other has E_{2} term

H^{s}H^{r}(Hom_{ℤG}(P, Hom(X, M))) = \begin{cases}0 & for r > 0 \\ H_{G}^{s}(X, M) & for r = 0. \end{cases}

This follows since each X_{r} is a finitely generated projective ℤG-module so (since G is finite), Hom(X_{s}, M) is cohomologically trivial. Since the second spectral sequence collapses, we have a spectral sequence

H^{r}(G, H^{s}(X, M)) ⟹ H_{G}^{r+s}(X, M).

The above constructions may be repeated for the subgroup H using the same X and P, so according to the hypotheses of the Theorem it follows that res_{G→H;X} : H_{G}^{∗}(X, M) → H_{H}^{∗}(X, M) is an isomorphism. For, the universal coefficient theorem provides an exact sequence

0 → Ext_{ℤ}^{1}(H_{s−1}(X), M) → H^{s}(Hom(X, M)) → Hom(H_{s}(X), M) → 0

and it is easy to check that this is a G-sequence. By an application of the 5-lemma, it suffices to show that H^{r}(G, L) → H^{r}(H, L) is an isomorphism for L equal to Ext_{ℤ}^{1}(H_{s−1}(X), M) or Hom(H_{s}(X), M). By condition (b) in the Lemma, since M is ℤ-free, Hom(H_{s}(X), M) is a direct sum of copies of M. Similarly, Ext_{ℤ}^{1}(H_{s−1}(X), M) is a direct sum of modules of the form M/kM for various k, and another application of the 5-lemma shows that H^{r}(G, M/kM)) → H^{r}(H, M/kM) is an isomorphism. It follows that we have an isomorphism at the E_{2} level, and so, by the comparison theorem for spectral sequences, we have an isomorphism H_{G}^{∗}(X, M) → H_{H}^{∗}(X, M).

To complete the proof we shall show that res_{G→H;X} : H_{G}^{q}(X, M) → H_{H}^{q}(X, M) is not surjective. To see this, note that we may also define a corestriction map

cor_{H→G;X} : H_{H}^{∗}(X, M) → H_{G}^{∗}(X, M)

by cor(f) = ∑_{t∈G/H} t(f) ∈ Hom_{G}(X, M) for f ∈ Hom_{H}(X, M), and

cor_{H→G;X} ∘ res_{G→H;X} = (G : H) : H_{G}^{q}(X, M) → H_{G}^{q}(X, M).

Since Hom(X_{q}, M) is cohomologically trivial, it follows that every element of

Hom_{G}(X_{q}, M)

is of the form

cor_{1→G;X}(g) = cor_{H→G;X}(∑_{h∈H} hg),

and since every element in degree q is a cocycle, it follows that cor_{H→G;X} is surjective. If res_{G→H;X} were also surjective the same would be true of multiplication by (G : H). Hence, it will suffice to show that H_{G}^{q}(X, M) (which is a finitely generated abelian group) is not finite. To this end, consider the composition

H^{q}(X, M) ⟶^{cor_{1→G:X}} H_{G}^{q}(X, M) ⟶^{res_{G→1;X}} H^{q}(X, M)

which amounts to multiplication by ∑_{t∈G} t. Since the effect of the latter on M^{G} is just multiplication by |G| and since we have assumed M^{G} is free and non-trivial, it follows that the image of H_{G}^{q}(X, M) → H^{q}(X, M) contains an element of infinite order. Hence, H_{G}^{q}(X, M) is not finite.

References

1. D. J. Benson and J. F. Carlson, Complexity and multiple complexes, Math. Zeit. 195 (1987), 221–238.

2. , Projective resolutions and Poincaré duality complexes, Trans. Amer. Math. Soc. 132 (1994), 447–488.

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

4. , A generalization of the transfer map in the cohomology of groups, Trans. Amer. Math. Soc. 108 (1963), 54–65.

5. S. Jackowski, Group homomorphisms inducing isomorphisms of cohomology, Topology 17 (1978), 303–307.

Mathematical Institute, 24–29 St. Giles, Oxford OX1 3LB

Department of Mathematics, Northwestern University, Evanston, IL 60208