Good moring,


Let $\Omega$ be a domain in $\mathbb{C}^n$ and $S\subset\Omega$ an analytic subset of codimension 1. *What can we say about the cohomology group $H^1(\Omega\backslash S, \mathbb{Z})$? E.g, when $\Omega$ is a ball?*


Any help is appreciated. Thanks in advance.


Duc Anh



Cohomology of Complements by an analytic subset?
I am not sure about cohomology with coefficients in $\mathbb Z$, but if we replace $\mathbb Z$ by $\mathbb Q$ and if $\Omega$ is a ball, then Poincar\'e duality implies that $H^1(\Omega\setminus S,\mathbb Q)$ is dual to $H^{2n-1}\_c(\Omega\setminus S,\mathbb Q)$ (cohomology with compact support). Now the exact sequence
$$
H^{2n-2}\_c(\Omega)\to H^{2n-2}\_c(S)\to H^{2n-1}\_c(\Omega\setminus S)\to H^{2n-1}\_c(\Omega)
$$
shows that $H^{2n-1}\_c(\Omega\setminus S)\cong H^{2n-2}\_c(S)$; the latter is isomorphic to $H^{2n-2}\_c(S\_{\mathrm{smooth}})$, that is, to $\mathbb Q^N$, where $N$ is the number of components of $S$.


