Andreotti-Frankel Theorem
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Andreotti%E2%80%93Frankel Theorem

In mathematics, the Andreotti-Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), states that if is a smooth, complex affine variety of complex dimension or, more generally, if is any Stein manifold of dimension , then admits a Morse function with critical points of index at most n, and so is homotopy equivalent to a CW complex of real dimension at most n.

Consequently, if is a closed connected complex submanifold of complex dimension , then has the homotopy type of a CW complex of real dimension . Therefore


This theorem applies in particular to any smooth, complex affine variety of dimension .


  • Andreotti, Aldo; Frankel, Theodore (1959), "The Lefschetz theorem on hyperplane sections", Annals of Mathematics, Second Series, 69: 713-717, doi:10.2307/1970034, ISSN 0003-486X, JSTOR 1970034, MR 0177422
  • Milnor, John W. (1963). Morse theory. Annals of Mathematics Studies, No. 51. Notes by Michael Spivak and Robert Wells. Princeton, NJ: Princeton University Press. ISBN 0-691-08008-9. Chapter 7.

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