Equivariant $A$-theory

TL;DR

This paper introduces a new construction of equivariant A-theory for group actions, producing infinite loop G-spaces and refined fixed point splittings, advancing the understanding of equivariant algebraic K-theory.

Contribution

It provides a novel construction of equivariant A-theory using spectral Mackey functors, including a second refinement with fixed point splittings, for finite group actions.

Findings

Produces an equivariant lift of Waldhausen's A-theory functor.

Shows fixed points correspond to bivariant A-theory of certain fibrations.

Introduces a second equivariant refinement with tom Dieck type splittings.

Abstract

We give a new construction of the equivariant $K$-theory of group actions (cf. Barwick et al.), producing an infinite loop $G$-space for each Waldhausen category with $G$-action, for a finite group $G$. On the category $R(X)$ of retractive spaces over a $G$-space $X$, this produces an equivariant lift of Waldhausen's functor $A(X)$, and we show that the $H$-fixed points are the bivariant $A$-theory of the fibration $X_{hH}\to BH$. We then use the framework of spectral Mackey functors to produce a second equivariant refinement $A_G(X)$ whose fixed points have tom Dieck type splittings. We expect this second definition to be suitable for an equivariant generalization of the parametrized $h$-cobordism theorem.