The signature package on Witt spaces
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 2, p. 241-310

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 * (BΓ)K * (C r * Γ) is injective.

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 * (BΓ)K * (C r * Γ) soit injective.

DOI : https://doi.org/10.24033/asens.2165
Classification:  35S35,  19K56,  58J20
Keywords: stratified pseudomanifold, Witt condition, iterated conic metrics, signature operator, index class, higher signatures, stratified homotopy invariance, assembly map
@article{ASENS_2012_4_45_2_241_0,
     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},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 45},
     number = {2},
     year = {2012},
     pages = {241-310},
     doi = {10.24033/asens.2165},
     zbl = {1260.58012},
     mrnumber = {2977620},
     language = {en},
     url = {http://www.numdam.org/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, Serie 4, Volume 45 (2012) no. 2, pp. 241-310. doi : 10.24033/asens.2165. http://www.numdam.org/item/ASENS_2012_4_45_2_241_0/

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