Homotopy Actions, Cyclic Maps and their Duals

Details Homotopy Actions, Cyclic Maps and their Duals Homotopy Actions, Cyclic Maps and their Duals

2005-09-23

arxiv.org/abs/math/0509568v1

Homology, Homotopy and Applications, vol 7(1) (2005), 169-184

Mathematics

Algebraic Topology

16 pages, LaTeX 2e

Scientific paper

An action of A on X is a map F: AxX to X such that F|_X = id: X to X. The restriction F|_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 <= 1. We also briefly discuss the relationship between a map being cyclic and its cocategory being <= 1.

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