{c}
{title2=$X_{hG}=EG\times_GX$}

= Homotopy quotient
{synonym}

For a <group action> on $X$ and a contractible free $G$-space $EG$, the diagonal orbit space $EG\times_GX$ maps to the <classifying space> $BG$ with fibre $X$. It defines <equivariant cohomology>. When $X\to X/G$ is a principal $G$-bundle, the map $EG\times_GX\to X/G$ has contractible fibre $EG$ and gives a <homotopy equivalence> for spaces of CW type. Freeness without the bundle hypotheses should not be substituted for this assertion for arbitrary topological groups.


 Borel construction (source code)