# Andreotti–Frankel theorem

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

Consequently, if {displaystyle Vsubseteq mathbb {C} ^{r}} is a closed connected complex submanifold of complex dimension {displaystyle n} , then {displaystyle V} has the homotopy type of a CW complex of real dimension {displaystyle leq n} . Therefore {displaystyle H^{i}(V;mathbb {Z} )=0,{text{ for }}i>n} and {displaystyle H_{i}(V;mathbb {Z} )=0,{text{ for }}i>n.} This theorem applies in particular to any smooth, complex affine variety of dimension {displaystyle n} .

References 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. This topology-related article is a stub. You can help Wikipedia by expanding it.

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