Equivariant K-theory and Resolution I: Abelian actions

TL;DR

This paper develops a geometric framework for understanding equivariant K-theory and cohomology of abelian group actions on manifolds, using resolutions and fibrations to establish an Atiyah-Hirzebruch isomorphism.

Contribution

It introduces a new resolution-based model for equivariant K-theory and cohomology for abelian actions, providing a direct proof of the equivariant Atiyah-Hirzebruch isomorphism.

Findings

Resolution of group actions yields fixed isotropy types.

Equivariant K-theory described via bundles over the base.

Model for delocalized equivariant cohomology using twisted forms.

Abstract

The smooth action of a compact Lie group on a compact manifold can be resolved to an iterated space, as made explicit by Pierre Albin and the second author. On the resolution the lifted action has fixed isotropy type, in an iterated sense, with connecting fibrations and this structure descends to a resolution of the quotient. For an abelian group action the equivariant K-theory can then be described in terms of bundles over the base with morphisms covering the connecting maps. A similar model is given, in terms of appropriately twisted deRham forms over the base as an iterated space, for delocalized equivariant cohomology in the sense of Baum, Brylinski and MacPherson. This approach allows a direct proof of their equivariant version of the Atiyah-Hirzebruch isomorphism.