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Equivariant evaluation subgroups and Rhodes groups. (English) Zbl 1125.55008

For the higher Rhodes groups \(\sigma _{n}(X,x_{0},G)\) and \(\tau _{n}=\tau _{n}(X,x_{0})\) (the \(n\)-th torus homotopy group), \(G\) a finite group acting freely on a compactly generated Hausdorff path connected space X with a basepoint \(x_{0}\), it holds that \(\sigma _{n}(X,x_{0},G)\rightarrow \sigma _{n}(X/G,p(x_{0}))\) is an isomorphism for all \(n\geq 1\).
In this paper the authors introduce new equivariant Gottlieb groups as subgroups of the higher Rhodes groups and discuss some basic properties such as “A space \(X\) is Gottlieb if and only if it is a Gottlieb-Fox space.”
Relationships among various evaluation subgroups are discussed. In particular, \(n\)-Gottlieb and equivariant \(n\)-Gottlieb spaces are related as follows: (1) for \(n\geq 2\) if X is equivariant \(n\)-Gottlieb then \(X/G\) is \(n\)-Gottlieb. (2) for \(n\geq 1\) if \(X/G\) is \(n\)-Gottlieb then \(X\) is equivariant \(n\)-Gottlieb (3) Suppose \(X\) is a finite aspherical \(G-CW\) space. If \(X~\)is equivariant 1-Gottlieb then \(X/G\) is 1-Gottlieb.

MSC:

55Q91 Equivariant homotopy groups
55Q05 Homotopy groups, general; sets of homotopy classes
55Q15 Whitehead products and generalizations
55M20 Fixed points and coincidences in algebraic topology
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References:

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