On the Nilpotency of Subgroups of Self-Homotopy Equivalences

Abstract

If X is a topological space, we denote by ε(X) the set of homotopy classes of self-homotopy equivalences of X. Then ε(X) is a group with group operation given by composition of homotopy classes. The group ε(X) is a natural object in homotopy theory and has been studied extensively—see [Ar] for a survey of known results and applications of ε(X). In this paper we continue our investigation of ε #(X), the subgroup of ε(X) consisting of homotopy classes which induce the identity on homotopy groups, and, to a lesser extent, of ε *#(X), the subgroup of ε #(X) consisting of homotopy classes which also induce the identity on homology groups (see §2 for precise definitions), which was begun in [A-L]. These groups are nilpotent and we focus primarily on the nilpotency class of ε #(X). The determination of this nilpotency class appears in the list of problems on ε(X) in [Ka, Problem 10]. For rational spaces we obtain both general results on the nilpotency class and a complete determination of the nilpotency class in specific cases. This leads to a lower bound for the nilpotency class of the groups ε #(X) for certain finite complexes X by using derationalization techniques.KeywordsMinimal ModelNilpotent GroupHomotopy ClassFinite OrderNilpotency ClassThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.