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March 25, 2015 By

Notes on Vanishing Homology

C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (4) 2009, pp. 118–126

December 30, 2009

Guillaume Valette, Instytut Mathematyczny PAN, ul. Sw. Tomasza 30, 31-027 Krakow, Poland; email: gvalette@impan.pl

Abstract/Résumé:

We introduce a homology theory devoted to the study of families such as semialgebraic or subanalytic families and in general of any family definable in an o-minimal structure. This also enables us to derive local metric invariants for germs of definable sets. The idea is to study the cycles which are vanishing when we approach a special fiber. We compute these groups and prove that they are finitely generated.

On introduit une théorie d’homologie pour les familles semialgébriques, sous-analytiques et plus généralement pour toute famille définissable dans une structure o-minimale. Cela permet aussi de définir des invariants locaux pour les singulariés définissables. L’idée est de considérer les cycles s’evanouissant lorsque l’on approche une fibre donnée. On calcule ces groupes et prouve qu’ils sont de type fini.

Keywords: definable sets, homology theory, metric invariants
AMS Subject Classification: Semialgebraic sets and related spaces 14P10

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