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Annales scientifiques de l'ENS - Parutions - série 4, 45 (2012)

Parutions < série 4, 45

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 45, fascicule 2 (2012)

Pierre Albin, Éric Leichtnam, Rafe Mazzeo, Paolo Piazza
The signature package on Witt spaces
Annales scientifiques de l'ENS 45, fascicule 2 (2012), 241-310

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Résumé :
Le forfait signature pour les espaces de Witt
Dans cet article nous prouvons plusieurs résultats pour l'opérateur de la signature sur un espace de Witt X compact orienté quelconque. Nous construisons une paramétrix de l'opérateur de la signature de X en raisonnant par récurrence sur la profondeur de X et en utilisant une analyse très fine de l'opérateur normal (près d'une strate). Ceci nous permet de montrer que le domaine maximal de l'opérateur de la signature est compactement inclus dans l'espace L^2 correspondant. On peut alors (re)démontrer que l'opérateur de la signature est essentiellement self-adjoint et a un spectre L^2 discret de multiplicité finie de sorte que son indice est bien défini. Nous donnons donc une nouvelle démonstration de certains résultats dus à Jeff Cheeger. Nous considérons ensuite le cas où X est muni d'un revêtement galoisien de groupe . Nous utilisons alors nos constructions pour définir la classe d'indice de signature analytique à valeurs dans le groupe de K-théorie K_*(C^*_r ). Nous généralisons dans cette situation singulière la plupart des résultats connus dans le cas où X est lisse. C'est ce qu'on appelle le forfait signature. En particulier, nous prouvons un nouveau théorème, purement topologique, qui permet de prouver l'invariance par homotopie stratifiée des hautes signatures de X (définies à l'aide de la L-classe homologique de X) pourvu que l'application d'assemblement rationnelle K_* (B ) Q K_*(C^*_r ) Q soit injective.

Mots-clefs : Pseudo-variétés stratifiées, condition de Witt, métriques coniques itérées, opérateur de signature, classe d'indice, hautes signatures, invariance par homotopie stratifiée, application d'assemblement

Abstract:
In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the `depth' of the singularity, is then used to show that the signature operator is essentially self-adjoint and has discrete spectrum of finite multiplicity, so that its index—the analytic signature of X—is well-defined. This provides an alternate approach to some well-known results due to Cheeger. We then prove some new results. By coupling this parametrix construction to a C^*_r Mishchenko bundle associated to any Galois covering of X with covering group , we prove analogues of the same analytic results, from which it follows that one may define an analytic signature index class as an element of the K-theory of C^*_r. We go on to establish in this setting and for this class the full range of conclusions which sometimes goes by the name of the signature package. In particular, we prove a new and purely topological theorem, asserting the stratified homotopy invariance of the higher signatures of X, defined through the homology L-class of X, whenever the rational assembly map K_* (B)K_*(C^*_r ) is injective.

Keywords: Stratified pseudomanifold, Witt condition, iterated conic metrics, signature operator, index class, higher signatures, stratified homotopy invariance, assembly map

Class. math. : 35S35; 19K56, 58J20


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

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