Cocyclic morphisms and dual G-sequences

Lee, Kee Young; Woo, Moo Ha

doi:10.1016/s0166-8641(00)00081-x

Cited by 4 publications

(2 citation statements)

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“…So in the coaction setting, we do not have the alternative point of view that a "coevaluation map" would provide. For results on cocyclic maps, see [Var69,HV75,Lim87,LW01]. In the case in which B = K(π, n), the set of cocyclic maps G (X, B) has been called the nth dual Gottlieb group of X (cf.…”

Section: Where ⊕ In [σω(A × X) X] Is Induced By the Suspension Strucmentioning

confidence: 99%

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Homotopy actions, cyclic maps and their duals

Arkowitz

Lupton

2005

Homology, Homotopy and Applications

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An action of A on X is a map F : A × X → X such that F | X = id : X → X. The restriction F | A : 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 of A on X have cyclic maps f and f with Ωf = Ωf , then ΩF and ΩF give the same action of ΩA on ΩX. 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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Section: Where ⊕ In [σω(A × X) X] Is Induced By the Suspension Strucmentioning

confidence: 99%

“…For the former, see [Gan67a,Zab76,IO03]. For the latter, see [Var69,HV75,LW01]. The Gottlieb groups have also received much attention in rational homotopy, following the results of Félix-Halperin in [FH82].…”

Section: Introductionmentioning

confidence: 99%