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2005 Homotopy actions, cyclic maps and their duals
Martin Arkowitz, Gregory Lupton
Homology Homotopy Appl. 7(1): 169-184 (2005).


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 $\le 1$.


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Martin Arkowitz. Gregory Lupton. "Homotopy actions, cyclic maps and their duals." Homology Homotopy Appl. 7 (1) 169 - 184, 2005.


Published: 2005
First available in Project Euclid: 13 February 2006

zbMATH: 1083.55011
MathSciNet: MR2155523

Primary: 55Q10
Secondary: 55M30, 55P30

Rights: Copyright © 2005 International Press of Boston


Vol.7 • No. 1 • 2005
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