Cohomology of Semidirect Product Groups

Abstract

We give examples of semidirect product groups G⋉A such that the HochschildSerre spectral sequence H(G,HA) ⇒ H(G⋉A) for Z/p-cohomology has nonzero differentials. Until now, few such examples have been known, especially when the normal subgroup A is abelian. In particular, Benson and Feshbach [2] mentioned that in all known semidirect products with A abelian, the spectral sequence satisfies: (1) All differentials after d2 are 0. (2) All differentials are 0 if A = (S1)n. (To be consistent with the notation for discrete groups A, HA here means the cohomology of the classifying space of A.) (3) All differentials are 0 if A = (Z/2)n and we consider cohomology with Z/2 coefficients. We give examples to show that all three statements can fail. In fact, there can be nonzero differentials at dp or later in all of these cases. I expect that there can be nonzero differentials arbitrarily far along in the spectral sequence in all of these cases, but the problem remains open. (For semidirect products G ⋉ A with A not abelian, Benson and Feshbach [2] gave examples of nonzero differentials arbitrarily far along in the spectral sequence for Z/2-cohomology.) It turns out that there is a very general reason why there will be nonzero differentials in some examples. If X is a G-space, then H(G,C(X)) admits Steenrod operations compatible with those on HG because it is the cohomology of the space (X × EG)/G, whereas there is no reason for H(G,M) to have Steenrod operations for a general G-module M . Thus Steenrod operations provide a fundamental obstruction for a G-module to be the representation of G on the cohomology of a G-space, as G. Carlsson found [4]; there is a useful exposition by Benson and Habegger [3]. If a semidirect product G ⋉ A has the G-action on A given by the dual of such a G-module, we can show that there must be nonzero differentials in the Hochschild-Serre spectral sequence. It is interesting to contrast these examples with Nakaoka’s theorem that the Hochschild-Serre spectral sequence has no differentials for any wreath product G⋉ Hn ([6], p. 50). Here G and H are any finite groups and G acts on Hn through a permutation representation G → Sn. It would be good to characterize algebraically the class of G-modules M over Z/p, say for a p-group G, such that the semidirect product G ⋉M has no differentials in the spectral sequence: it seems to be fairly close to the class of permutation modules, but there are some other interesting examples. I would like to thank J. L. Alperin, Steve Siegel, and Peter Sin for many conversations on group cohomology. Steve Siegel read an early version of this paper and made useful suggestions. Also, this work was partially supported by the NSF.