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

Abstract

For a based space X, we consider the group $¥mathscr{E}$_#n(X) of all self homotopy classes α of X such that α_# = id:π_i(X) → π_i(X), for all i≤n, where n≤∞, and the group $¥mathscr{E}$_Ω(X) of all α such that Ωα = id. Analogously, we study the semigroups $¥mathscr{Z}$_#n(X) and $¥mathscr{Z}$_Ω(X) defined by replacing ‘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.