Definition
Let \(G\) be a topological group of dimension \(k\). The \(i\)th dimensional \(G\)-equivariant chains \(C^G_i(X)\) of a \(G\)-space \(X\) consist of \(G\)-equivariant maps \(C\to X\), where \(C\) is a compact “reasonable” (possibly singular subanalytic subset of \(\mathbb R^n\)) \(i+k\) dimensional space with a free \(G\)-action. The boundary map is the usual boundary map.
Then the cohomology of the complex \(C^G_\bullet(X)\) is isomorphic to the equivariant cohomology \(H^i_G(X)\) defined via the Borel construction.