Equivariant loops on classifying spaces

TL;DR

This paper computes the homology of equivariant loop spaces on classifying spaces of simplicial monoids with anti-involution, extending known theorems and analyzing fixed points in algebraic K-theory contexts.

Contribution

It introduces a method to compute homology of equivariant loops on classifying spaces with anti-involution, generalizing the group completion theorem.

Findings

Homology of equivariant loops computed under certain conditions.

Homology of $C_2$-fixed points in algebraic K-theory modeled and analyzed.

Fixed point space sometimes group complete, but not always.

Abstract

We compute the homology of the space of equivariant loops on the classifying space of a simplicial monoid $M$ with anti-involution, provided $\pi_0 (M)$ 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 compute the homology of the $C_2$-fixed points of a Segal-type model of the algebraic $K$-theory of an additive category with duality. As an application we show that this fixed point space is sometimes group complete, but not in general.