#### Volume 20, issue 5 (2020)

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Equivariant loops on classifying spaces

### Kristian Jonsson Moi

Algebraic & Geometric Topology 20 (2020) 2511–2552
##### Abstract

We compute the homology of the space of equivariant loops on the classifying space of a simplicial monoid $M$ with anti-involution, provided ${\pi }_{0}\left(M\right)$ is central in the homology ring of $M\phantom{\rule{-0.17em}{0ex}}$. The proof is similar to McDuff and Segal’s proof of the group completion theorem. Then we give an analogous computation of the homology of the ${C}_{2}$–fixed points of a $\Gamma$–space-type delooping of an additive category with duality with respect to the sign circle. As an application we show that this fixed-point space is sometimes group complete, but in general not.

##### Keywords
algebraic topology, loop space, classifying space, group completion
##### Mathematical Subject Classification 2010
Primary: 55P35, 55P48, 55P91