## Homotopy actions, cyclic maps and their duals.(English)Zbl 1083.55011

In homotopy theory, the analogue of a topological group action on a (pointed) space is the homotopy action of a pointed space on another one: $$F: (A,a_0)\times (X,x_0)\to (X,x_0)$$ such that the restricton map $$F|_{\{a_0\}\times X}= \text{id}_X$$. Then the restriction map $$F|_{A\times\{x_0\}}: A\to X$$ is called a cyclic map. Action and cyclic maps have been well-studied in the literature.
In this paper the authors prove some general results about actions and cyclic maps. In particular, they prove that two actions loop to the same action. Their results dualize. The dual of a cyclic map is called a cocyclic map. They clarify the relation between cocyclic maps and the connecting maps of cofibration sequence.

### MSC:

 55Q05 Homotopy groups, general; sets of homotopy classes 55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) 55P30 Eckmann-Hilton duality

### Keywords:

homotopy action; Gottlieb group
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