Thanks to Alexander duality, we know that for each compact subset $K$ of $\mathbb{C}$ there is an isomorphism $$H_1(\mathbb{C} \backslash K) \simeq \prod_{i \in CC(K)} \mathbb{Z},$$ where $CC(K)$ is the set of all connected components of $K$. It seems clear to me that this isomorphism is given by the index of the closed path in $\mathbb{C}\backslash K$. More formally, to a cohomological class $[\gamma]$, we associate the family $\{Ind_{z_i} \gamma\}_{i \in CC(K)}$ where the $z_i$ are arbitrary points in each connected component. How can I proove that this application is actually the previous isomorphism given by Alexander Duality. Is there any book in which this is explained ?

Furthermore, I'd like to obtain such an isomorphism with an open set $\Omega$ containing $K$. Is it possible to characterize $H^1(\Omega \backslash K)$ with the indexes of the paths ?

Thank you.

  • $\begingroup$ You may use the long exact sequence of the pair $(\Omega ,K)$ : $H_1 (K)\rightarrow H_1 (\Omega ) \rightarrow H_1 (\Omega\backslash K )\rightarrow H_0 (K) \rightarrow H_0 (\Omega )$. If \Omega is e.g. simply connected then you get $0\rightarrow H_1 (X\backslash K )\rightarrow H_0 (K) \rightarrow {\mathbb C}$, so that you may relate $H_1 (X\backslash K )$ with the set of connected components of $K$. But if $\Omega$ is not simply connected, this is no longer true. $\endgroup$ – Paul Broussous Nov 13 '15 at 11:20

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