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I learned about Tate cohomology that for finite group $G$ and $G$-module $A$, if $\hat{H}^0(H,A)=\hat{H}^1(H,A)=0$ for any subgroup $H \subset G$, then $\hat{H}^k(G,A)=0$ for all $k$.

Is there similar statement for topological space? For example, does it hold for topological space $X$, if $H^0(Y,\mathbb{Z})=\mathbb{Z}, H^1(Y,\mathbb{Z})=0$ for any open $Y \subset X$, then $H^k(X,\mathbb{Z})=0$ for $k \geq 2$?

aerile
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1 Answers1

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There are a lot disconnected opens. And for these opens zero cohomology cannot be $\mathbb Z$ . So, for hausdorf space this condition means, that your space is point.

And even if this open is connected there is a problem. Assume that your CW complex contains D^n as open subset. Then you always have open subset with nontrivial first homology.

Galois group
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