Homology of subset of orbit space

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.

• I am not sure what kind of an answer your are looking for. Consider for instance the action of $Z_2$ on $X=A=S^k$, $n\ne k$, such that the generator of $Z_2$ acts as the antipodal map. topospaces.subwiki.org/wiki/Homology_of_real_projective_space – Moishe Kohan Dec 30 '18 at 23:33
• I am especially interested in the case when $G$ is perfect – piotrmizerka Dec 31 '18 at 10:11
• Also, I would like to consider rather unproper inclusions $A\subset X$. – piotrmizerka Dec 31 '18 at 10:14
• One can easily modify my examples to satisfy your two conditions. In any case, you should edit your question to reflect what you are actually interested in. – Moishe Kohan Dec 31 '18 at 16:27

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
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 $$X$$.