Let $M$ be a smooth $G-$manifold, where $G$ is a compact Lie group. From the result of S.Illman that $M$ admits an equivariant triangulation. Moreover, we can construct the equivariant singular homology $H^{G}_{*}(M;k)$ where $k$ is a covariant coefficient system for $G$ ovr a ring $R$. For every standart equivariant $n-$simplex $\triangle_{n}(K_{0},...,K_{n})$ and $G-$map $$T:\triangle_{n}(K_{0},...,K_{n})\longrightarrow M$$ the induced map by the $G-$action is $$u:\triangle_{n}\longrightarrow M/G.$$ What is the relation between the equivariant singular homology group $H^{G}_{*}(M;k)$ and the homology group $H_{*}(M/G;R)$ ?

Cohomology of Groupsfor this construction). – Chris Gerig Jun 13 '13 at 4:54