On the equivarant de-Rham cohomology for non-compact Lie groups

TL;DR

This paper establishes an equivalence between the existence of equivariant extensions of closed forms in the Cartan model and in the homotopy quotient for non-compact Lie group actions, advancing understanding of equivariant de-Rham cohomology.

Contribution

It introduces a new equivalence result connecting Cartan model extensions and homotopy quotient extensions for non-compact Lie groups.

Findings

Equivalence between Cartan model and homotopy quotient extensions.

Applicable to non-compact Lie group actions.

Provides a new perspective on equivariant de-Rham cohomology.

Abstract

Let $G$ be a connected and non-necessarily compact Lie group acting on a connected manifold $M$. In this short note we announce the following result: for a $G$-invariant closed differential form on $M$, the existence of a closed equivariant extension in the Cartan model for equivariant cohomology is equivalent to the existence of an extension in the homotopy quotient.