Equivariant de Rham Theory and Graphs

TL;DR

This paper demonstrates that certain theorems in equivariant de Rham theory can be translated into graph theory, simplifying their proofs and computations.

Contribution

It shows that some fundamental theorems in equivariant de Rham theory are equivalent to graph-theoretic results, leveraging recent advances in the field.

Findings

Equivariant cohomology computations can be reduced to graph theory.

Some classical theorems in equivariant de Rham theory are equivalent to graph-theoretic statements.

The approach simplifies the understanding and calculation of equivariant cohomology.

Abstract

Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be theorems about graphs. In this paper we show that for some familiar theorems, this is indeed the case.