Equivariant loops on classifying spaces

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 $M$. 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 $\mathrm{\Gamma }$–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.