14P10 — 3 articles found.

Note on Poincaré $L^p$ Type Inequality for Differential Forms on Semialgebraic Sets

C. R. Math. Rep. Acad. Sci. Canada Vol. 34 (1) 2012, pp. 23–32
Vol.34 (1) 2012
Leonid Shartser Details
(Received: 2011-06-02 )
(Received: 2011-06-02 )

Leonid Shartser, Department of Mathematics, University of Toronto, 40 St. George St., Toronto, ON M5S 2E4; e-mail: shartl@math.toronto.edu

Abstract/Résumé:

We study local and global Poincaré type \(L^p\) inequalities on a compact semialgebraic subset of \(\mathbb{R}^n\) for \(p\gg 1\). As a consequence, we obtain an isomorphism between \(L^p\) cohomology and singular cohomology of a normal compact semialgebraic set. The global inequality is derived from the local one, while the local inequality is proved by means of a semialgebraic Lipschitz deformation retraction with estimates on its derivatives.

On étudie les inégalités locales et globales de type \(L^p\) de Poincaré sur un sous-ensemble compact semialgébrique de \(\mathbb{R}^n\) pour \(p\gg 1\). Par conséquent, nous obtenons un isomorphisme entre la cohomologie \(L^p\) et la cohomologie singulière d’un ensemble normal compact semialgébrique. L’inégalité globale est dérivée de la locale, tandis que l’inégalité locale est prouvée au moyen d’une rétraction de déformation semialgébrique Lipschitz avec des estimations sur ses dérivés.

Keywords:

AMS Subject Classification: Semialgebraic sets and related spaces 14P10

PDF(click to download): Note on Poincaré $L^p$ Type Inequality for Differential Forms on Semialgebraic Sets

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
Vol.32 (1) 2010
Leonid Shartser; Guillaume Valette Details
(Received: 2008-09-26 , Revised: 2009-10-29 )
(Received: 2008-09-26 , Revised: 2009-10-29 )

Leonid Shartser, Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON M5S 2E4; e-mail: shartl@math.toronto.edu

Guillaume Valette, Instytut Matematyczny PAN, 31-027 Krakow, Poland; email: gvalette@impan.pl

Abstract/Résumé:

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.

Keywords: De Rham theory, definable sets, homology theory, metric invariants

AMS Subject Classification: Semialgebraic sets and related spaces 14P10

PDF(click to download): A De Rham Theorem for $L^infty$ Forms and Homology on Singular Spaces

Notes on Vanishing Homology

C. R. Math. Rep. Acad. Sci. Canada Vol. 31 (4) 2009, pp. 118–126
Vol.31 (4) 2009
Guillaume Valette Details
(Received: 2009-06-18 )
(Received: 2009-06-18 )

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

PDF(click to download): Notes on Vanishing Homology

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