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Annales scientifiques de l'ENS - Parutions - série 4, 49 (2016)

Parutions < série 4, 49

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 49, fascicule 2 (2016)

Paulo Carrillo Rouse, Bai-Ling Wang
Geometric Baum-Connes assembly map for twisted Differentiable Stacks
Annales scientifiques de l'ENS 49, fascicule 2 (2016), 277-323

Télécharger cet article : Fichier PDF

Résumé :
Le morphisme d'assemblage géométrique de Baum-Connes pour des Champs différentiables tordus
Nous construisons le morphisme d'assemblage géométrique de Baum-Connes pour des groupoïdes de Lie tordus, à savoir des groupoïdes de Lie avec un PU(H)-fibré principal équivariant. La construction est basée sur l'utilisation des groupoïdes de déformation, ces objets permettent en particulier de donner une construction géométrique des morphismes shriek associés et d'établir la fonctorialité. Les principaux résultats de cet article sont la définition des groupes de K-homologie géométrique tordue et la construction du morphisme d'assemblage. Même dans le cas non tordu le fait que les groupes de K-homologie géométrique et le morphisme d'assemblage (géométrique) pour des groupoïdes de Lie sont bien définis est nouveau ; en effet, ceci a été esquissé par Connes dans son livre pour des groupoïdes de Lie générales sans aucune restriction, en particulier pour des groupoïdes non séparés. Nous montrons aussi l'invariance par Morita du morphisme d'assemblage, donnant ainsi un sens précis au morphisme d'assemblage géométrique de Baum-Connes pour des champs différentiables tordus. Nous discutons la relation de notre morphisme d'assemblage avec le morphisme associé à la S^1-extension centrale. La relation avec le morphisme analytique est traitée, ainsi que quelques cas où il y a isomorphisme. Un outil important est le morphisme de Thom tordu dans le cas équivariant par rapport à un groupoïde que nous établissons dans l'appendice.

Mots-clefs : K-théorie tordue, théorie de l'indice, groupoïdes de Lie et champs différentiables.

Abstract:
We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant PU(H)-principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the assembly map. Even in the untwisted case the fact that the geometric K-homology groups and the geometric assembly map are well defined for Lie groupoids is new, as it was only sketched by Connes in his book for general Lie groupoids without any restrictive hypothesis, in particular for non Hausdorff Lie groupoids. We also prove the Morita invariance of the assembly map, giving thus a precise meaning to the geometric assembly map for twisted differentiable stacks. We discuss the relation of the assembly map with the associated assembly map of the S^1-central extension. The relation with the analytic assembly map is treated, as well as some cases in which we have an isomorphism. One important tool is the twisted Thom isomorphism in the groupoid equivariant case which we establish in the appendix.

Keywords: Twisted K-theory, Index theory, Lie groupoids and Differentiable stacks. MSC.

Class. math. : 58J22; 19K35,19K56,46L80.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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