For a based space , we consider the group of all self homotopy classes of such that , for all , where , and the group of all such that . Analogously, we study the semigroups and defined by replacing '' by '' above. There is a chain of containments of the -groups and the -semigroups, and we discuss examples for which the containment is proper. We then obtain various conditions on which ensure that the -groups and the -semigroups are equal. When 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 -groups and -groups of certain low dimensional Lie groups.
"Homotopy classes of self-maps and induced homomorphisms of homotopy groups." J. Math. Soc. Japan 58 (2) 401 - 418, April, 2006. https://doi.org/10.2969/jmsj/1149166782