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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 $S^{n-1}$? $\endgroup$ Oct 16, 2015 at 9:20
  • $\begingroup$ 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 Du
    Oct 17, 2015 at 16:04
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    $\begingroup$ 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. $\endgroup$ Oct 17, 2015 at 16:52
  • $\begingroup$ What about the case of free action? $\endgroup$
    – Ryan Du
    Jul 13, 2016 at 16:07

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