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Homology of semi-pfaffian sets. (Homologie des ensembles semi-pfaffiens.) (French) Zbl 0853.32004

Summary: A pfaffian subset of an open semianalytic subset \(M\) of \(\mathbb{R}^n\) is a finite intersection of relatively compact semianalytic sets of \(\mathbb{R}^n\) and non spiraling leaves of analytic codimension 1 foliations of \(M.\) The class of semipfaffian subsets of \(M\) is the smallest collection of subsets of \(M\) containing the pfaffian subsets of \(M,\) which is stable under finite intersection, finite union and complement in \(M\). The class of \(T\)-pfaffian sets is the smallest collection of subsets of \(\mathbb{R}^n,\) containing the pfaffian sets, which is stable under finite intersection, finite union, topological closure and linear projection. We prove the finiteness of Betti numbers of relatively compact semipfaffian sets and the finiteness of the number of connected components of \(T\)-pfaffian sets.

MSC:

32C05 Real-analytic manifolds, real-analytic spaces
32C25 Analytic subsets and submanifolds
58A99 General theory of differentiable manifolds
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