Homology of subset of orbit space

Question

Assume a finite group $G$ acts on a topological space $X$ and $A\subseteq X$. Denote by $q$ the quotient map from $X$ to the orbit space $X/G$ (we take the quotient topology). Moreover, let $H_n(A)=0$ for some $n\geq 1$.

Let us consider a subset $q(A)$ of $X/G$ with the induced subset topology.

When is it the case that $H_n(q(A))=0$? (I mean especially group properties of $G$) If the answer is not possible in general, it would be nice to point out specific situations too (e.g. $X$ - manifold, smooth manifold, Lie group, etc.)

The analogous problem for cohomology is also interesting to me.

Answer 1

Since you are asking about finite perfect groups, here is an example. Let $G=A_5$. Then $H_2(G)\cong {\mathbb Z}_2\ne 0$ (see here: Schur multiplier is another name for $H_2$) This group acts freely on $$ X= S^5\times S^7, $$ see Theorem 1.1 of

Fixity and free group actions on products of spheres, by A.Adem, J.Davis and O.Unlu, Commentarii Mathematici Helvetici, 2004, Vol. 79, pp 758--778.

It follows that $H_2(X/G)\cong H_2(G)\ne 0$, while $H_2(X)=0$.

If you want $A$ a proper subset of $X$, take $X$ to be the product $$ S^5\times S^7 \times S^{2019} $$ and let $G$ act trivially on $S^{2019}$. Then take $A$ to be the product $$ S^5\times S^7 \times\{p\}\subset X. $$

On the other hand, there are, of course, examples where $q(A)$ is acyclic. For instance, let $A$ be a point. Or let $X$ be a smooth manifold, $G\times X\to X$ a smooth action, $A$ a small $G$-invariant ball containing a fixed point of the $G$-action on $X$.