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Usually one defines the traditionnal singular homology (let's say in $\mathbb{Z}$ and on a topological space $X$) by using singular $p$-chains. A singular $p$-chain is a finite formal sum $\sum c_{\sigma} \sigma$ where $\sigma$ are $p$-simplexes in $X$ and $c_{\sigma}$ integers. A well-known theorem, usually proven with barycentric decomposition is the Mayer-Vietoris theorem.

Theorem : Let $\{U,V\}$ be an open covering of $X$. Then on has the following long exact sequence $$\cdots \rightarrow H_{n+1}(X) \xrightarrow\, H_{n}(U\cap V) \xrightarrow\, H_{n}(U) \oplus H_{n}(V) \xrightarrow\, H_{n}(X) \xrightarrow\, H_{n-1} (U\cap V) \rightarrow \cdots$$

Now my question is the following one. Instead of defining usual singular homology, one can define locally finite singular homology, noted $H^{lf}_{\cdot}(X)$ by using infinite formal sums $\sum c_{\sigma} \sigma$ but which are locally finite. That means that for all $x$ their is a neighbourhood $V_x$ which meets only a finite number of $supp(\sigma)$. Does Mayer-Vietoris hold in this context ? I.e, do we have a long exact sequence $$\cdots \rightarrow H^{lf}_{n+1}(X) \xrightarrow\, H^{lf}_{n}(U\cap V) \xrightarrow\, H^{lf}_{n}(U) \oplus H^{lf}_{n}(V) \xrightarrow\, H^{lf}_{n}(X) \xrightarrow\, H^{lf}_{n-1} (U\cap V) \rightarrow \cdots$$ for any open covering $\{U,V\}$ of $X$ ? This seems not easy to me because the barycentric decomposition will probably not work as in the finite case.

Any help or recommandation for a book that works with locally finite homology will be much appreciated.

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No, in case of $H^{lf}_\bullet$ Mayer-Vietoris doesn't hold.

For locally compact $X$ it is well-known that $H^{lf}_n(X)=\widetilde H_n(\overline X)$, where $\overline X$ is one-point compactification. To construct counterexample take $X=S^3$, let $U$ and $V$ be $S^3\setminus x$ and $S^3\setminus y$ for $x\ne y$. As it easy to see, $\overline{U\cap V}=S^3\vee S^1$, so $H^{lf}_1(U\cap V)=\mathbb Z$, and $\overline{U}=\overline{V}=S^3$.

Finally, consider the fragment of desired sequence $$ \ldots\to H^{lf}_2(X)\to H^{lf}_1(U\cap V) \to H^{lf}_1(U)\oplus H^{lf}_1(V)\to\ldots $$ which, in light of the above, becomes $$ \ldots\to0\to\mathbb Z\to0\to\ldots $$

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  • $\begingroup$ Thank you for your answer ! Can you provide a reference for the "well-known fact" ? Or a book which works with such things. I must add that actually the topological space I'm interested in is $\mathbb{C}^*$. Is there any chance that Mayer-Vietoris works in this context ? $\endgroup$
    – C. Dubussy
    Commented Mar 15, 2016 at 20:04
  • $\begingroup$ @C.Dubussy, for example, see here $\endgroup$ Commented Mar 17, 2016 at 13:52
  • $\begingroup$ Ok thank you. Last remark. In your counter-example, shouldn't we look the $H_3$ instead of the $H_1$ ? $\endgroup$
    – C. Dubussy
    Commented Mar 17, 2016 at 14:25
  • $\begingroup$ @C.Dubussy, i has added the arguments that i mean in the answer $\endgroup$ Commented Mar 17, 2016 at 15:08

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