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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Sign up to join this communityLet $G$ 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?
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.