A group $G$ acts freely on a manifold $M$, then $H^*_G(M)=H^*(M/G)$.

>Why is $H^*_G(M)$ a torsion $H^*_G$-module, where $H^*_G=H^*_G(pt)=H^*(BG)$?
>
>If $G=T=(S^1)^{n+1}$ is a torus then $H^*_G=H^*_T=\mathbb{Q}[t_0,...,t_n]$. Why does $t_i$ act on $H^*_G(M)$ by multiplication by $0$?

More importantly, I would like to understand the intuition behind these.

For example: does the answer to the second question above have to do with the infinitesimal action of the Lie algebra of $T$ on $M$? what does it mean to say that the non-torsion part of $H^*_G(M)$ is contributed by the $G$-fixed part of M?


  [1]: http://math.stackexchange.com/questions/1878385/basic-question-intuition-for-equivariant-cohomology