Cohomology of Group Extensions

Introduction. Let G be a group, K an invariant subgroup of G. The purpose of this paper is to investigate the relations between the cohomology groups of G, K, and G/K. As in the case of fibre spaces, it turns out that such relations can be expressed by a spectral sequence whose term E2 is HiG/K, HiK)) and whose term Em is the graduated group associated with i7(G). This problem was first studied by R. C. Lyndon in his thesis [12]. Lyndon's procedure was to replace the full cochain complex of G by an equivalent bigraduated subcomplex (of "normal" cochains, in his sense). His main result (generalized from the case of a direct product to the case of an arbitrary group extension, according to his indications) is that the bigraduated group associated with if(G) is isomorphic with a factor group of a subgroup of HiG/K, HiK)). His methods can also be applied to special situations, like those considered in our Chapter III, and can give essentially the same results. We give here two different approaches to the problem. In Chapter I we carry out the method sketched by one of us in [13]. This method is based on the Cartan-Leray spectral sequence, [3; l], and can be generalized to other algebraic situations, as will be shown in a forthcoming paper of Cartan-Eilenberg [2]. Since the details of the Cartan-Leray technique have not been published (other than in seminar notes of limited circulation), we develop them in Chapter I. The auxiliary theorems we need for this purpose are useful also in other connections. In Chapter II, which is independent of Chapter I, we obtain a spectral sequence quite directly by filtering the group of cochains for G. This filtration leads to the same group E2 = HiG/K, HiK)) (although we do not know whether or not the succeeding terms are isomorphic to those of the first spectral sequence) and lends itself more readily to applications, because one can identify the maps which arise from it. This is not always the case with the first filtration, and it is for this reason that we have developed the direct method in spite of the somewhat lengthy computations which are needed for its proofs. Chapter III gives some applications of the spectral sequence of Chapter II. Most of the results could be obtained in the same manner with the spectral sequence of Chapter I. A notable exception is the connection with the theory of simple algebras which we discuss in §5. Finally, let us remark that the methods and results of this paper can be transferred to Lie Algebras. We intend to take up this subject in a later paper.