Let $X$ be an affine variety of dimension $n$ over $\mathbb{C}$.
Does the analytic space associated with $X$ have the homotopy type of a $n$-dimensional CW complex?
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1$\begingroup$ see paper of Milnor , maths.ed.ac.uk/~aar/papers/milnorcw.pdf $\endgroup$– user21574Commented Jan 24, 2017 at 20:41
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1 Answer
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If $X$ is smooth, and if you ask to have the same homotopy type of a CW complex of real dimension at most $n$, this is precisely the statement of the Andreotti-Frankel theorem.
It is true, more generally, for a Stein manifold of complex dimension $n$.
If $X$ is arbitrarily singular, the same theorem holds, provided $X$ is irreducible. This was shown by Karčjauskas for algebraic varieties and by Hamm for Stein spaces.
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5$\begingroup$ When $X$ is any algebraic variety (in particular, any affine variety ), the aforementioned CW-compex is also finite. This implies that the fundamental group $\pi_1(X)$ and the homology and cohomology groups $H_i(X, \, \mathbb{Z})$, $H^i(X, \, \mathbb{Z})$ are all finitely generated. This is not true anymore fror an arbitrary smooth Stein space, for which the CW-complex can be infinite. Take for instance $$X=\{x \in \mathbb{C}\, | \, \sin x \neq 0 \}.$$ $\endgroup$ Commented Jan 20, 2017 at 16:01