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Comptes Rendus Mathématique
Volume 336, n° 9
pages 763-768 (mai 2003)
Doi : 10.1016/S1631-073X(03)00168-7
Received : 14 January 2003 ;  accepted : 11 Mars 2003
Virtual Betti numbers of real algebraic varieties

Clint  McCrory a ,  Adam  Parusinacute;ski b
aMathematics Department, University of Georgia, Athens, GA 30602, USA 
bDépartement de mathématiques, Université d'Angers, 2, bd Lavoisier, 49045 Angers cedex, France 

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We show that for all   the  -th mod 2 Betti number of compact nonsingular real algebraic varieties has a unique extension to a virtual Betti number   defined for all real algebraic varieties, such that if   is a closed subvariety of   then  . We show by example that there is no natural weight filtration on the  -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. Parusinacute;ski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

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On montre que pour tout entier positif   le  -ième nombre de Betti de la cohomologie à coefficients dans   des variétés algébriques réelles compactes nonsingulières admet une unique extension en un nombre de Betti virtuel  , défini pour toute variété algébrique réelle, telle que pour une sous-variété fermée  ,  . On donne un exemple qui montre qu'il n'existe pas de filtration par le poids naturelle sur la cohomologie à coefficients dans   des variétés algébriques réelles telle que les nombres de Betti virtuels soient les caractéristiques d'Euler par le poids associées à cette filtration. Pour citer cet article : C. McCrory, A. Parusinacute;ski, C. R. Acad. Sci. Paris, Ser. I 336 (2003).




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