On G-spaces, Serre classes and G-nilpotency

Abstract

0. Introduction. The concept of a nilpotent action on a nilpotent group was studied in (2), and was crucial to the definition of a nilpotent map (or nilpotent fibration) in (4). These ideas had, of course, already been treated systematically in (1)†. The concept found a new, and related, application to homotopy theory in (3), where the following situation was studied. Let X be a nilpotent space and let G be a group which ‘acts’on X. Here it is not strictly speaking necessary to specify the nature of the action, provided that there is an induced action of G on the homotopy and homology groups of X. Thus G might be a group of base-point preserving homeomorphisms of X, or a group of (based) homotopy classes of self-homotopy-equivalences of X. We then discussed in (3) questions related to the nilpotency of the induced actions of G on the homotopy and homology groups of X and also relativized the theory to a discussion of fibrations on which the group G acts; to obtain interesting results related to nilpotency, it turned out to be at least necessary to suppose the fibration F → E → B quasinilpotent (7), meaning that all spaces are connected and π1B acts nilpotently on the homology groups of F.