Filtrations of global equivariant K-theory

TL;DR

This paper extends filtrations of algebraic K-theory to a global equivariant setting, providing new computational tools and revealing similarities with Burnside ring filtrations, thus advancing understanding of equivariant homotopy groups.

Contribution

It introduces global equivariant filtrations of K-theory spectra, generalizing previous constructions and simplifying the computation of algebraic filtrations on representation rings.

Findings

Representation ring filtrations are easier to express globally.

Filtrations on Burnside rings relate to symmetric products of spheres.

The work generalizes Arone and Lesh's filtrations to the equivariant context.

Abstract

Arone and Lesh constructed and studied spectrum level filtrations that interpolate between connective (topological or algebraic) K-theory and the Eilenberg-MacLane spectrum for the integers. In this paper we consider (global) equivariant generalizations of these filtrations and of another closely related class of filtrations, the modified rank filtrations of the K-theory spectra themselves. We lift Arone and Lesh's description of the filtration subquotients to the equivariant context and apply it to compute algebraic filtrations on representation rings that arise on equivariant homotopy groups. It turns out that these representation ring filtrations are considerably easier to express globally than over a fixed compact Lie group. Furthermore, they have formal similarities to the filtration on Burnside rings induced by the symmetric products of spheres, which was computed by Schwede.