Invariants under deformation of the actions of topological groups

TL;DR

This paper demonstrates that the equivariant cohomology of a space remains invariant under homotopic deformations of the group action, establishing isomorphisms between cohomologies for different but homotopic actions.

Contribution

It introduces a method to associate objects in equivariant derived categories under homotopic actions, proving invariance of equivariant cohomology for such deformations.

Findings

Equivariant cohomology is invariant under homotopic actions.

Constructs a correspondence between objects in equivariant derived categories.

Shows isomorphism of cohomologies for Lie groups acting on subanalytic spaces.

Abstract

Let $\varphi$ and $\varphi'$ be two homotopic actions of the topological group $G$ on the topological space $X$. To an object $A$ in the $G$-equivariant derived category $D_{\varphi}(X)$ of $X$ relative to the action $\varphi$ we associate an object $A'$ of category $D_{\varphi'}(X)$, such that the corresponding $G$-equivariant compactly supported cohomologies $H_{G,c}(X,\,A)$ and $H_{G,c}(X,\,A')$ are isomorphic. When $G$ is a Lie group and $X$ is a subanalytic space, we prove that the $G$-equivariant cohomologies $H_{G}(X,\,A)$ and $H_{G}(X,\,A')$ are also isomorphic.