Representatives of the Thom class of a vector bundle

TL;DR

This paper reviews methods for constructing equivariant cohomology classes and introduces a new family of representatives for the universal Thom class, including a simple, symmetric explicit representative, contrasting with previous known representatives.

Contribution

It applies a method by Berline, Getzler, and Vergne to produce a new family of Thom class representatives, revealing a surprising discrepancy with Mathaï and Quillen's representative.

Findings

Introduces a new family of Thom class representatives

Constructs an explicit, simple, symmetric representative

Shows the new family does not include Mathaï and Quillen's representative

Abstract

After a review of several methods designed to produce equivariant cohomology classes, we apply one introduced by Berline, Getzler and Vergne, to get a family of representatives of the universal Thom class of a vector bundle. Surprisingly, this family does not contain the representative given by Matha\"{\i} and Quillen. However it contains a particularly simple and symmetric representative that we construct explicitly.