A linearization map for genuine equivariant algebraic $K$-theory

TL;DR

This paper develops a new genuine equivariant algebraic K-theory for group actions, enabling the study of classical invariants in an equivariant setting through a novel linearization map.

Contribution

It introduces a genuine G-spectrum valued algebraic K-theory for coefficient systems of rings, connecting it with equivariant A-theory via a linearization map.

Findings

Constructed a genuine G-spectrum K_G(Z[π_1(X)]) for G-spaces.

Established a comparison with equivariant A-theory.

Provided a framework for equivariant classical invariants.

Abstract

We introduce a version of algebraic $K$-theory for coefficient systems of rings which is valued in genuine $G$-spectra for a finite group $G$. We use this construction to build a genuine $G$-spectrum $K_G(\mathbb{Z}[\underline{\pi_1(X)}])$ associated to a $G$-space $X$, which provides a home for equivariant versions of classical invariants like the Wall finiteness obstruction and Whitehead torsion. We provide a comparison between our $K$-theory spectrum and the equivariant $A$-theory of Malkiewich--Merling via a genuine equivariant linearization map.