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Virtual Betti numbers of real algebraic varieties. (English. Abridged French version) Zbl 1073.14071
For a real algebraic variety \(X\) the authors define virtual Betti numbers \(\beta_i(X)\). More precisely, the authors prove that, for any nonnegative integer \(i\), there exists a unique extension of the \(i\)th modulo 2 Betti number of compact nonsingular real algebraic varieties to a virtual Betti number \(\beta_i\) defined for all real algebraic varieties and verifying the following property: if \(Y\) is a closed subvariety of a real algebraic variety \(X\), then \(\beta_i(X)= \beta_i(Y) + \beta_i(X\setminus Y)\). The article also contains an example which shows that there is no natural weight filtration on the \(\mathbb{Z}/2\)-cohomology (with compact supports) of real algebraic varieties such that the virtual Betti numbers are the weighted Euler characteristics.

14P05 Real algebraic sets
14P25 Topology of real algebraic varieties
14F45 Topological properties in algebraic geometry
Full Text: DOI
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