Virtual Betti numbers of real algebraic varieties

By C. McCrory, A. Parusiński. 2002

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Abstract We show that for all i⩾0 the i-th mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti number βi defined for all real algebraic varieties, such that if Y is a closed subvariety of X then βi(X)=βi(X⧹Y)+βi(Y). We show by example that there is no natural weight filtration on the Z 2 -cohomology of real algebraic varieties with compact supports such that the virtual Betti numbers are the weighted Euler characteristics. To cite this article: C. McCrory, A. Parusinski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

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