Eta and rho invariants on manifolds with edges

Piazza, Paolo; Vertman, Boris

doi:10.5802/aif.3287

Eta and rho invariants on manifolds with edges
[Invariants êta and rho sur variétés stratifiée de profondeur 1]

Piazza, Paolo ¹ ; Vertman, Boris ²

¹ Sapienza Università di Roma Dipartimento di Matematica Roma (Italy)
² University of Oldenburg Institute of Mathematics Oldenburg (Germany)

Annales de l'Institut Fourier, Tome 69 (2019) no. 5, pp. 1955-2035.

Résumé
Abstract

Dans cet article, nous démontrons l’existence des invariants êta et des invariants rho de Atiyah–Patodi–Singer et Cheeger–Gromov pour une classe d’opérateurs de type Dirac sur des variétés stratifiées de profondeur 1 munies d’une métrique incomplète de type edge. Notre analyse s’applique, en particulier, à l’opérateur de signature et à l’opérateur de Dirac sur une variété spin. Nous établissons aussi des théorèmes de Atiyah–Patodi–Singer sur des variétés stratifiées de profondeur 1 avec bord et sur leurs revêtements de Galois. Nos arguments s’appuyent sur l’analyse microlocale du développement asymptotique du noyau de la chaleur pour un laplacien de Dirac associé à une métrique incomplète de type edge. Nous donnons des applications de cette analyse à l’étude des propriétés de stabilité des invariants rho que nous avons définit.

We establish existence of eta-invariants as well as of the Atiyah–Patodi–Singer and the Cheeger–Gromov rho-invariants for a class of Dirac operators on an incomplete edge space. Our analysis applies in particular to the signature and the spin Dirac operator. We derive an analogue of the Atiyah–Patodi–Singer index theorem for incomplete edge spaces and their non-compact infinite Galois coverings with edge singular boundary. Our arguments are based on the microlocal analysis of the heat kernel asymptotics associated to the Dirac laplacian of an incomplete edge metric. As an application, we discuss stability results for the two rho-invariants we have defined.

Reçu le : 2016-05-06
Révisé le : 2018-05-16
Accepté le : 2018-10-24
Publié le : 2019-09-16

DOI : 10.5802/aif.3287

Classification : 58J20, 58J28
Keywords: stratified space, incomplete edge metrics, spin manifold, signature operator, spin Dirac operator, heat kernel asymptotic, eta invariant, rho invariant, Fredholm index
Mot clés : espace stratifiée, métrique incomplète de type edge, varieté spin, opérateur de signature, opérateur spin-Dirac, développement asymptotique du noyau de la chaleur, invariants êta, invariants rho, indice de Fredholm

Affiliations des auteurs :

Piazza, Paolo ¹ ; Vertman, Boris ²

¹ Sapienza Università di Roma Dipartimento di Matematica Roma (Italy)
² University of Oldenburg Institute of Mathematics Oldenburg (Germany)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Piazza, Paolo; Vertman, Boris. Eta and rho invariants on manifolds with edges. Annales de l'Institut Fourier, Tome 69 (2019) no. 5, pp. 1955-2035. doi : 10.5802/aif.3287. https://aif.centre-mersenne.org/articles/10.5802/aif.3287/

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