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Let $X$ be a $CW$-complex and let $G$ be a group acting "cellularly" on $X$, that is, for each $g\in G$, the induced homeomorphism $\phi_g:X\to X$ takes cells to cells.

Also assume that the action is free.

Then it seems to me that the orbit space $X/G$ also becomes a CW-complex.

Question. Are there any results knows which relate the homologies of $X/G$ with the homology of $X$, taking into account the action of $G$?

In general, are there any results known which relate the homologies of the orbit space of a $G$-space $X$, where now $X$ is just any topological space and not necesarily a CW-complex.

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    I bumped into similar matters some time ago, and I was pointed to Bredon's book on group actions on spaces, *Introduction to Compact Transformation Groups*. – 2017-01-14
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    Thank you. I checked the book out. There is a chapter on "Homological Theory of Finite Group Actions" but it does not seem to address the question I asked in my post. Can you tell where in the book should I look at. – 2017-01-14
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    The quotient need not be a CW complex, I think, unless you assume in addition that $G$ is a Lie group. As for homology, the relation is by no means easy (it depends on the action!), you should ask a more specific question. Even in the case of principal G-bundles (I think this is what you are after) you need a Serre spectral sequence to relate the homology groups! – 2017-01-15

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