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.)



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



Artin vanishing for Stein manifolds and restriction maps
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.


