A Note on the Nilpotency of Subgroups of Self-Homotopy Equivalences

Abstract

Let X be a space that has the homotopy type of a finite simply connected CW complex. We denote by E(X) the group of homotopy classes of self-homotopy equivalences of X. This group has been extensively studied (see [1] for a survey). In this paper we consider the subgroup En#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on the homotopy groups πi(X) for i⩽n, and the subgroup E#(X) consisting of homotopy classes of self-homotopy equivalences of X that induce the identity on all the homotopy groups. Our first result is as follows. 1991 Mathematics Subject Classification 55P62, 55P10.