I tried to find the homology groups of the quotient of the unit sphere $S^{n-1}$ by an action of a finite subgroup $G$ of $SO(n)$. I'm especially concerned with $$H_i(S^{n-1}/G),\quad 1\leq i\leq n-2.$$ I haven't found any reference for this problem.
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$\begingroup$ Are you assuming that $G$ acts freely on $S^{n-1}$? $\endgroup$– Neil StricklandCommented Oct 16, 2015 at 9:20
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$\begingroup$ No, the action need not to be free. $G<SO(n)$ acts on $R^n$ by linear transformations. It preserves the unit sphere $S^{n-1}$. $\endgroup$– Ryan DuCommented Oct 17, 2015 at 16:04
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1$\begingroup$ OK. The situation for free actions is well understood, but I do not know of anything systematic than can be said for non-free actions. $\endgroup$– Neil StricklandCommented Oct 17, 2015 at 16:52
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$\begingroup$ What about the case of free action? $\endgroup$– Ryan DuCommented Jul 13, 2016 at 16:07
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