Homotopy classes of self-maps and induced homomorphisms of homotopy groups

For a based space X , we consider the group \mathscr{E}_{\# n}(X) of all self homotopy classes \alpha of X such that \alpha_{\#} = \mathrm{id}:\pi_i(X) \to \pi_i(X) , for all i\le n , where n \le \infty , and the group \mathscr{E}_{\Omega}(X) of all \alpha such that \Omega \alpha = \mathrm{id} . Analogously, we study the semigroups \mathscr{Z}_{\# n}(X) and \mathscr{Z}_{\varOmega}(X) defined by replacing ' \mathrm{id} ' by ' 0 ' above. There is a chain of containments of the \mathscr{E} -groups and the \mathscr{Z} -semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on X which ensure that the \mathscr{E} -groups and the \mathscr{Z} -semigroups are equal. When X is a group-like space, we derive lower bounds for the order of these groups and their localizations. In the last section we make specific calculations for the \mathscr{E} -groups and \mathscr{Z} -groups of certain low dimensional Lie groups.