Suppose $G$ is a compact topological group acting on an $m$-homology manifold $M$ over some ring $R$ by homeomorphisms.
Assume that the action of $G$ is effectively finite on a closed $G$-invariant subspace $A$ such that $A$ does not contain a connected component of $M$.
(Recall that an effectively finite action means that the quotient group of $G$ by the ineffective kernel $K$ is finite. In the case I am interested, one can assume $G/K \cong \mathbb{Z}/k$)
Does it follow that $H^{m}(A/G,R)=0$?
