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In the setting of complex Stein manifolds $X$ of complex dimension $d$, the theorem of Andreotti--Frankel implies the vanishing of the singular cohomology group $H^i(X,\mathbb Z)=0$ for $i>d$. With complex coefficients, a simple argument for this is to compute the cohomology in terms of the cohomology of the de Rham complex. Their theorem gives a more precise Morse-theoretic statement.

Now let $U\subset X$ be another Stein manifold open in $X$, and assume that the map $\mathcal O(X)\to \mathcal O(U)$ has dense image. (This condition is not automatic, and necessary for the following.)

Conjecture. In top degree, the map $H^d(X,\mathbb Z)\to H^d(U,\mathbb Z)$ is surjective.

Is this known? The same result should also be true with constructible coefficients. With $\mathbb C$-coefficients, it follows from the comparison with de Rham cohomology (at least when cohomology groups are finite-dimensional, which I'm happy to assume). Is there some argument using Morse theory?

The motivation for the question is that the analogue in rigid-analytic geometry is true (but I found it quite surprising); it is essentially equivalent to a version of Artin vanishing for affine schemes over absolutely integrally closed valuation rings stated by Gabber in Oberwolfach last year.

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    $\begingroup$ @JasonStarr An example would be a small open ball inside affine space $\endgroup$
    – Peter Scholze
    Commented Sep 2, 2021 at 11:21
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    $\begingroup$ What topology are you using? For the open unit disk in the complex plane, the holomorphic restriction of $1/(1-z)$ is certainly not approximated by restrictions of entire holomorphic functions in the uniform topology (since these are bounded on the closure of the disk). $\endgroup$
    – Jason Starr
    Commented Sep 2, 2021 at 11:40
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    $\begingroup$ I'm using the Frechet topology on $\mathcal O(X)$ and $\mathcal O(U)$ (of uniform convergence on compact subsets), as I believe is the standard choice. (Sorry for not specifying.) $\endgroup$
    – Peter Scholze
    Commented Sep 2, 2021 at 11:42
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    $\begingroup$ Interesting question. The inclusion of $\mathbb{C}^*\subset \mathbb{C}$ is an open immersion of Stein manifolds where the map on $H^d$ is not surjective, but I'm not sure about the density condition for function spaces. $\endgroup$
    – Donu Arapura
    Commented Sep 2, 2021 at 12:53
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    $\begingroup$ It doesn't have dense image: One argument is to observe that the residue at $0$ of a function on $\mathbb C^\times$ gives a nonzero continuous functional which vanishes on the image of $\mathcal{O}(\mathbb C)$. The same argument applies more generally to hyperplane complements. $\endgroup$
    – Peter Scholze
    Commented Sep 2, 2021 at 12:56

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The pairs (U,X) are called Runge pairs. The homology version of your statement is proved in the paper of Andreotti and Narasimhan Annals of Math vol 76 no 3 (1962) 499-509 using Morse Theory.The title of the paper is "A Topological property of Runge pairs" The paper by Coltoiu Mihalache titled On the Homology Groups of Stein spaces and Runge pairs, Journal fur reine und angewandte Mathematik volume 371 no 5 pages 215-220 proves the homology statement for Runge pairs of Stein spaces.

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  • $\begingroup$ Thanks a lot for these references! (Is the version with constructible coefficients also known?) $\endgroup$
    – Peter Scholze
    Commented Sep 2, 2021 at 19:26
  • $\begingroup$ It was my pleasure. $\endgroup$
    – Mohan Ramachandran
    Commented Sep 2, 2021 at 19:27
  • $\begingroup$ I don't think the case of constructible coefficients is known but I am not sure. $\endgroup$
    – Mohan Ramachandran
    Commented Sep 2, 2021 at 19:35

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