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Let $G$ be a compact and totally disconnected group acting on a paracompact space $X$.

Does the orbit map $X \rightarrow X/G$ induce an isomorphism (or a monomorphism) in Alexander-Spanier cohomology with closed support?

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The following is a related result of Bredon et al and Lowen. You may find more in this book by Bredon.

Theorem. Let $G$ be a totally disconnected compact group that acts on a locally compact Hausdorff space $X$, and let $k$ be a field of characteristic $0$. Then the orbit projection $X\rightarrow X/G$ induces an isomorphism $$H_c^{*}(X/G;k)\cong \text{Fix}(G;H_c^{*}(X;k)).$$

Also, take a look at this paper by Satea Deo.

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  • $\begingroup$ It is true for locally compact spaces. I dont know that it is true for paracompact spaces. I read Deo's article $\endgroup$
    – Mehmet Onat
    Commented Dec 22, 2016 at 11:58

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