## Bulletin of the Belgian Mathematical Society - Simon Stevin

### Realizing homotopy group actions

#### 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}. }

#### Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 21, Number 4 (2014), 685-710.

Dates
First available in Project Euclid: 23 October 2014

https://projecteuclid.org/euclid.bbms/1414091009

Digital Object Identifier
doi:10.36045/bbms/1414091009

Mathematical Reviews number (MathSciNet)
MR3271327

Zentralblatt MATH identifier
1305.55007

#### Citation

Blanc, David; Sen, Debasis. Realizing homotopy group actions. Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 4, 685--710. doi:10.36045/bbms/1414091009. https://projecteuclid.org/euclid.bbms/1414091009