# 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 Dec 30, 2018 at 23:33
• I am especially interested in the case when $G$ is perfect Dec 31, 2018 at 10:11
• Also, I would like to consider rather unproper inclusions $A\subset X$. Dec 31, 2018 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. Dec 31, 2018 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 $$X$$.