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

Abstract

For a based space $X$, we consider the group ${ℰ}_{\#n}\left(X\right)$ of all self homotopy classes $\alpha$ of $X$ such that ${\alpha }_{\#}=\mathrm{i}\mathrm{d}:{\pi }_i\left(X\right)\to {\pi }_i\left(X\right)$, for all $i\le n$, where $n\le \mathrm{\infty }$, and the group ${ℰ}_{\mathrm{\Omega }}\left(X\right)$ of all $\alpha$ such that $\mathrm{\Omega }\alpha =\mathrm{i}\mathrm{d}$. Analogously, we study the semigroups ${𝒵}_{\#n}\left(X\right)$ and ${𝒵}_{\mathrm{\Omega }}\left(X\right)$ defined by replacing '$\mathrm{i}\mathrm{d}$' by '$0$' above. There is a chain of containments of the $ℰ$-groups and the $𝒵$-semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on $X$ which ensure that the $ℰ$-groups and the $𝒵$-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 $ℰ$-groups and $𝒵$-groups of certain low dimensional Lie groups.