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# Homology, Homotopy and Applications

## Volume 18 (2016)

### Number 1

### Equivariant $\Gamma$-spaces

Pages: 295 – 324

DOI: http://dx.doi.org/10.4310/HHA.2016.v18.n1.a16

#### Author

#### Abstract

The aim of this note is to provide a comprehensive treatment of the homotopy theory of $\Gamma \textrm{-}G$-spaces for $G$ a finite group. We introduce two level and stable model structures on $\Gamma \textrm{-}G$-spaces and exhibit Quillen adjunctions to $G$-symmetric spectra with respect to a flat level and a stable flat model structure, respectively. Then we give a proof that $\Gamma \textrm{-}G$-spaces model connective equivariant stable homotopy theory along the lines of the proof in the non-equivariant setting given by Bousfield and Friedlander. Furthermore, we study the smash product of $\Gamma \textrm{-}G$-spaces and show that the functor from $\Gamma \textrm{-}G$-spaces to G-symmetric spectra commutes with the derived smash product. Finally, we show that there is a good notion of geometric fixed points for $\Gamma \textrm{-}G$-spaces.

#### Keywords

equivariant homotopy theory, infinite loop spaces

#### 2010 Mathematics Subject Classification

55P47, 55P91

Published 31 May 2016