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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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##### References:
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