The signature package on Witt spaces

Albin, Pierre; Leichtnam, Éric; Mazzeo, Rafe; Piazza, Paolo

The signature package on Witt spaces
[Le forfait signature pour les espaces de Witt]

Albin, Pierre ; Leichtnam, Éric ; Mazzeo, Rafe ; Piazza, Paolo

Annales scientifiques de l'École Normale Supérieure, Tome 45 (2012), p. 241-310 / Harvested from Numdam

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Abstract

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 $Γ$ . Nous utilisons alors nos constructions pour définir la classe d’indice de signature analytique à valeurs dans le groupe de $K$ -théorie $K_{*} (C_{r}^{*} Γ)$ . 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 $K_{*} (B Γ) \otimes ℚ \to K_{*} (C_{r}^{*} Γ) \otimes ℚ$ soit injective.

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 $C_{r}^{*} Γ$ Mishchenko bundle associated to any Galois covering of $X$ with covering group $Γ$ , 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 $C_{r}^{*} Γ$ . 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 $K_{*} (B Γ) \otimes ℚ \to K_{*} (C_{r}^{*} Γ) \otimes ℚ$ is injective.

Publié le : 2012-01-01

MR 2977620 | Zbl 1260.58012 | 2 citations dans Numdam

DOI : https://doi.org/10.24033/asens.2165
Classification: 35S35, 19K56, 58J20
Mots clés: pseudo-variétés stratifiées, condition de Witt, métriques coniques itérées, opérateur de signature, classe d'indice, hautes signatures, invariance par homotopie stratifiée, application d'assemblement

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     author = {Albin, Pierre and Leichtnam, \'Eric and Mazzeo, Rafe and Piazza, Paolo},
     title = {The signature package on Witt spaces},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     volume = {45},
     year = {2012},
     pages = {241-310},
     doi = {10.24033/asens.2165},
     mrnumber = {2977620},
     zbl = {1260.58012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ASENS_2012_4_45_2_241_0}
}

Albin, Pierre; Leichtnam, Éric; Mazzeo, Rafe; Piazza, Paolo. The signature package on Witt spaces. Annales scientifiques de l'École Normale Supérieure, Tome 45 (2012) pp. 241-310. doi : 10.24033/asens.2165. http://gdmltest.u-ga.fr/item/ASENS_2012_4_45_2_241_0/

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