Delocalized equivariant cohomology and resolution

TL;DR

This paper develops a canonical resolution process for smooth group actions on manifolds, tracking equivariant cohomology theories through the resolution to produce simplified models and an explicit equivariant Chern character.

Contribution

It introduces a resolution method for smooth group actions that preserves and tracks equivariant cohomology theories, leading to explicit models and a Chern character construction.

Findings

Resolved models for equivariant K-theory and cohomology are constructed.

The resolution induces fibrations on boundary faces, simplifying the structure.

An explicit equivariant Chern character is derived over the resolved quotient.

Abstract

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established by the authors in the context of manifolds with corners; the canonical construction induces fibrations on the boundary faces of the resolution resulting in an `equivariant resolution structure'. Here, equivariant K-theory and the Cartan model for equivariant cohomology are tracked under the resolution procedure as is the delocalized equivariant cohomology of Baum, Brylinski and MacPherson. This leads to resolved models for each of these cohomology theories, in terms of relative objects over the resolution structure and hence to reduced models as flat-twisted relative objects over the resolution of the quotient. An explicit equivariant Chern character is then constructed, essentially as in the non-equivariant case, over the resolution of the quotient.