A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary

Paulo Carrillo Rouse (1) , Jean-Marie Lescure (2, 3, 4) , Bertrand Monthubert (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 LMBP - Laboratoire de Mathématiques Blaise Pascal
3 UCA [2017-2020] - Université Clermont Auvergne [2017-2020]
4 CNRS - Centre National de la Recherche Scientifique

We give a cohomological formula for the index of a fully elliptic pseudodifferential operator on a manifold with boundary. As in the classic case of Atiyah-Singer, we use an embedding into an euclidean space to express the index as the integral of a cohomology class depending in this case on a noncommutative symbol, the integral being over a $C^\infty$-manifold called the singular normal bundle associated to the embedding. The formula is based on a K-theoretical Atiyah-Patodi-Singer theorem for manifolds with boundary that is drawn from Connes' tangent groupoid approach.

Mots clés: index theory; groupoids; manifolds with boundary; K-theory

Domaines: Algèbres d'opérateurs [math.OA]; Géométrie différentielle [math.DG]; K-théorie et homologie [math.KT]

Paulo Carrillo Rouse, Jean-Marie Lescure, Bertrand Monthubert. A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary. Journal of Topology and Analysis, 2014, 6 (1), pp.27-74. ⟨10.1142/S1793525314500046⟩. ⟨hal-00717784v3⟩