Homotopy actions, cyclic maps and their duals

Abstract

An action of $A$ on $X$ is a map $F\colon A\times X \to X$ such that $F\vert_X = \id \colon X\to X$. The restriction $F\vert_A$: $A \to X$ 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 $F'$ of $A$ on $X$ have cyclic maps $f$ and $f'$ with $\Omega f = \Omega f'$, then $\Omega F$ and $\Omega F'$ give the same action of $\Omega A$ on $\Omega X$. 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 $\le 1$. We also briefly discuss the relationship between a map being cyclic and its cocategory being $\le 1$.