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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.

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