Comptes Rendus
no. 23-24
Geometry/Differential Topology
An index theorem for manifolds with boundary
[Un théorème d'indice pour des variétés à bord]

Présenté par : Alain Connes

Paulo Carrillo-Rouse1 ; Bertrand Monthubert1
1 Institut de mathématiques de Toulouse, université de Toulouse, 31062 Toulouse cedex 9, France
Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1393-1398.

Dans le livre Non Commutative Geometry, 1994, II.5, Connes donne une preuve du théorème de l'indice d'Atiyah–Singer pour des variétés fermées en utilisant des groupoïdes de déformation et des actions appropriées de ceux-ci dans RN. Nous suivons ces idées pour montrer un théorème d'indice pour des variétés à bord.

In Connes (Non Commutative Geometry, 1994, II.5), a proof is given of the Atiyah–Singer index theorem for closed manifolds by using deformation groupoids and appropriate actions of these on RN. Following these ideas, we prove an index theorem for manifolds with boundary.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.10.021
Paulo Carrillo-Rouse 1 ; Bertrand Monthubert 1

1 Institut de mathématiques de Toulouse, université de Toulouse, 31062 Toulouse cedex 9, France 
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Paulo Carrillo-Rouse; Bertrand Monthubert. An index theorem for manifolds with boundary. Comptes Rendus. Mathématique, Volume 347 (2009) no. 23-24, pp. 1393-1398. doi : 10.1016/j.crma.2009.10.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.10.021/

[1] P. Carrillo-Rouse A Schwartz type algebra for the tangent groupoid, K-theory and Noncommutative Geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 181-199

[2] A. Connes Non Commutative Geometry, Academic Press, Inc., San Diego, CA, 1994

[3] C. Debord; J.M. Lescure K-duality for pseudomanifolds with isolated singularities, J. Funct. Anal., Volume 219 (2005), pp. 109-133

[4] C. Debord; J.M. Lescure; V. Nistor Groupoids and an index theorem for conical pseudomanifolds, J. Reine Angew. Math., Volume 628 (2009), pp. 1-35

[5] R. Lauter; B. Monthubert; V. Nistor Pseudodifferential analysis on continuous family groupoids, Doc. Math., Volume 5 (2000), pp. 625-656

[6] R. Melrose The Atiyah–Patodi–Singer Index Theorem, Research Notes in Mathematics, vol. 4, A K Peters, Ltd., Wellesley, MA, 1993 xiv+377 pp. (English summary)

[7] I. Moerdijk; J. Mrčun Introduction to Foliations and Lie Groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, 2003

[8] B. Monthubert Groupoids and pseudodifferential calculus on manifolds with corners, J. Funct. Anal., Volume 199 (2003) no. I, pp. 243-286

[9] B. Monthubert; F. Pierrot Indice analytique et groupoïdes de Lie, C. R. Acad. Sci. Paris Sér. I, Volume 325 (1997), pp. 193-198

[10] B. Monthubert Contribution of noncommutative geometry to index theory on singular manifolds, Geometry and Topology of Manifolds, Banach Center Publ., vol. 76, Polish Acad. Sci., Warsaw, 2007, pp. 221-237

[11] A. Paterson Groupoids, Inverse Semigroups, and their Operator Algebras, Progress in Mathematics, vol. 170, Birkhäuser Boston, Inc., Boston, MA, 1999 (xvi+274 pp)

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