Homotopy actions, cyclic maps and their duals

Abstract

An action of A on X is a map F : A £ X ! X such that FjX = id: X ! X. The restriction FjA: A ! X of an action is called a cyclic map. Special cases of these notions include group actions and the Gottlieb groups of a space, each of which has been studied extensively. We prove some general results about actions and their Eckmann-Hilton duals. For instance, we classify the actions on an H-space that are compatible with the H-structure. As a corollary, we prove that if any two actions F and F 0 of A on X have cyclic maps f and f 0 with ›f = ›f 0 , then ›F and ›F 0 give the same action of ›A on ›X. We introduce a new notion of the category of a map g and prove that g is cocyclic if and only if the category is less than or equal to 1. From this we conclude that if g is cocyclic, then the Berstein-Ganea category of g is 6 1. We also briefly discuss the relationship between a map being cyclic and its cocategory being 6 1.