Fixed points of the equivariant algebraic $K$-theory of spaces

TL;DR

This paper proves a tom Dieck-type splitting for the fixed points of the equivariant algebraic K-theory spectrum of spaces with finite group actions, demonstrating compatibility with the equivariant suspension spectrum splitting.

Contribution

It establishes a fixed point splitting for the equivariant algebraic K-theory of spaces, extending the understanding of its structure and relations to suspension spectra.

Findings

Fixed points admit a tom Dieck-type splitting.

Splitting is compatible with the equivariant suspension spectrum.

Results align with independent findings by John Rognes.

Abstract

In a recent work Malkiewich and Merling proposed a definition of the equivariant $K$-theory of spaces for spaces equipped with an action of a finite group. We show that the fixed points of this spectrum admit a tom Dieck-type splitting. We also show that this splitting is compatible with the splitting of the equivariant suspension spectrum. The first of these results has been obtained independently by John Rognes.