Equivariant characteristic forms in the Cartan model and Borel equivariant cohomology

TL;DR

This paper demonstrates the compatibility of differential geometric and topological methods for constructing equivariant characteristic classes, extending known results from compact to non-compact Lie groups.

Contribution

It provides a proof of compatibility for non-compact Lie groups, generalizing previous results limited to compact connected groups, and introduces a differential geometric construction.

Findings

Compatibility established for compact Lie groups

Extension to non-compact Lie groups proposed

Framework applicable to broader classes of groups

Abstract

We show the compatibility of the differential geometric and the topological construction of equivariant characteristic classes for compact Lie groups. Our analysis motivates a differential geometric construction for equivariant characteristic classes in the non-compact case. This compatibility is generally assumed and used in various cases, but there is only a proof for compact connected Lie groups in the literature, see the work of Raoul Bott and Loring Tu. Our proof applies and generalizes ideas of Johan Dupont and Ezra Getzler.