I am aware of the existence of a transfer homomorphism in the setting of so called "regular $G$-complexes", as described e.g. in Bredon's Introduction to Compact Transformation Groups.
But suppose that I have a smooth action of a finite group $G$ on a compact manifold $M$. Supposing that I work with singular cohomology with coefficients in a field of characteristic $0$ or prime to the order of $G$, do I still have an isomorphism $H^*(M)^G = H^*(M/G)$?
Perhaps it is so that a compact $G$-manifold can always be equipped with a structure of a regular $G$-complex? If this is the case I'd be happy to learn a source where this is spelled out.