Equivariant Cohomology of homotopy equivalent spaces

Question

Let $V$ is contractible space with $T$ torus action, then can I say their equivariant cohomology (in Borel sense) are equal ? i.e for $\bullet = point$ , $H_T^*(V)=H_T^*(\bullet)$ ?

Answer 1

Equivariant cohomology are defined as ordinary cohomology of $V \times_G EG$. There is a projection $p: V \times_G EG \rightarrow BG$.This projection is fiber bundle with fiber $V$.

If $V$ is contractible, then this projection is homotopy equivalence, hence induces isomorphism on cohomology $H^{*} (BG) \rightarrow H^{*} ( V \times_G EG)$. I.e. isomorphism $H^*_G ( \bullet ) \rightarrow H^*_G(V)$