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definable sets — 2 results found.

      
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A De Rham Theorem for $L^\infty$ Forms and Homology on Singular Spaces
C. R. Math. Rep. Acad. Sci. Canada Vol. 32 (1) 2010, pp. 24–32
Leonid Shartser; Guillaume Valette (Received: 2008/09/26, Revised: 2009/10/29)

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We introduce a notion of a smooth \(L^{\infty}\) form on singular (semialgebraic) spaces \(X\) in \(\mathbb{R}^n\). An \(L^\infty\) form is the data of a stratification \(\Sigma\) of \(X\) and a collection of smooth forms \(\omega\) on the nonsingular strata with matching tangential components on the adjacent strata and bounded size (in the metric induced from \(\mathbb{R}^n\)). We prove Stokes’ Theorem and Poincaré’s Lemma for \(L^\infty\) forms. As a result we obtain a De Rham type theorem establishing a natural isomorphism between the singular cohomology and the cohomology of smooth \(L^{\infty}\) forms.

On introduit la notion d’une forme \(L^\infty\) pour des espaces singuliers semialgébriques. Une forme lisse \(L^\infty\) est la donnée d’une stratification et d’une famille de forme lisses sur les strates coincidant le long des strates adjacentes. On prouve la formule de Stokes et le lemme de Poincaré pour les formes \(L^\infty\). On en déduit un théorème de type De Rham établissant un isomorphisme naturel entre la cohomologie des formes \(L^\infty\) et la cohomologie singulière.

Notes on Vanishing Homology
C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (4) 2009, pp. 118–126
Guillaume Valette (Received: 2009/06/18)

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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.

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algebraic number theory approximation property automorphisms Bessel functions Boson-fermion correspondence C*-algebra Carmichael number center problem Chebyshev transform classification Classification of simple C*-algebras composition operators continued fractions Cuntz Semigroup elliptic curves fixed point Fourier transform function fields. functoriality general relativity generic property ideals indefinite inner product inductive limits of sub-homogeneous C*- algebras Irrational rotation algebra J-Hermitian matrix K-theory Kahler manifolds L-functions maximal ideal space nonexpansive mapping numerical range orthogonal polynomials Predual space prime number property SP Renormalization rotation algebras Salem number semi-reciprocal polynomials tracially approximate splitting interval algebras unbounded traces uniqueness Weak Markov set Whitney problems

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