Virtual Betti numbers of real algebraic varieties

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 numberβ_i defined for all real algebraic varieties, such that if Y is a closed subvariety of X then β_i(X)=β_i(X?Y)+β_i(Y). We show by example that there is no natural weight filtration on the $\text{Z}{}_2$-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. Parusiński, C. R. Acad. Sci. Paris, Ser. I 336 (2003).