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

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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:πi(X) → πi(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.

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