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Let $\varphi$ and $\varphi'$ be two homotopic actions of the topologicalgroup $G$ on the topological space $X$. To an object $A$ in the $G$-equivariantderived category $D_{\varphi}(X)$ of $X$ relative to the action $\varphi$ weassociate an object $A'$ of category $D_{\varphi'}(X)$, such that thecorresponding $G$-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 asubanalytic space, we prove that the $G$-equivariant cohomologies$H_{G}(X,\,A)$ and $H_{G}(X,\,A')$ are also isomorphic.

Invariants under deformation of the actions of topological groups