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october 2014 Realizing homotopy group actions
David Blanc, Debasis Sen
Bull. Belg. Math. Soc. Simon Stevin 21(4): 685-710 (october 2014). DOI: 10.36045/bbms/1414091009

Abstract

For any finite group $G$, we define the notion of a \emph{Bredon homotopy action} of $G$, modelled on the diagram of fixed point sets \ensuremath{(\mathbf{X}\sp{H})\sb{H\leq G}} for a $G$-space $\mathbf{X}$, together with a pointed homotopy action of the group \ensuremath{N\sb{G}H/H} on \ensuremath{\mathbf{X}\sp{H}/(\bigcup\sb{H<K} \mathbf{X}\sp{K}).} We then describe a procedure for constructing a suitable diagram \ensuremath{\underline{\mbox{X}}:{\EuScript O}_{G}\op\to{\EuScript Top}} from this data, by solving a sequence of elementary lifting problems. If successful, we obtain a $G$-space \ensuremath{\mathbf{X}'}\ realizing the given homotopy information, determined up to Bredon $G$-homotopy type. Such lifting methods may also be used to understand other homotopy questions about group actions, such as transferring a $G$-action along a map \ensuremath{f:\mathbf{X}\to \mathbf{Y}. }

Citation

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David Blanc. Debasis Sen. "Realizing homotopy group actions." Bull. Belg. Math. Soc. Simon Stevin 21 (4) 685 - 710, october 2014. https://doi.org/10.36045/bbms/1414091009

Information

Published: october 2014
First available in Project Euclid: 23 October 2014

zbMATH: 1305.55007
MathSciNet: MR3271327
Digital Object Identifier: 10.36045/bbms/1414091009

Subjects:
Primary: 55P91
Secondary: 55R35 , 55S35 , 58E40

Keywords: Bredon theory , equivariant homotopy type , group actions , homotopy actions , obstructions

Rights: Copyright © 2014 The Belgian Mathematical Society

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Vol.21 • No. 4 • october 2014
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