Singular spaces, groupoids and metrics of positive scalar curvature

2018/03/07 by Paolo Piazza, Vito Felice Zenobi · 1 citation
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Algebra over a field #Class (philosophy) #Computer science #Curvature #Dirac operator #Geometry #Mathematical analysis #Mathematics #Noncommutative and Quantum Gravity Theories #Operator theory #Order (exchange) #Pure mathematics #Scalar (mathematics) #Scalar curvature #Spin (aerodynamics) #math.DG #math.KT

paper · pdf · doi:10.1016/j.geomphys.2018.09.016

50 pages

arxiv created 2018/03/07 · openalex publication_date 2018/12/05 · arxiv updated 2019/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We define and study, under suitable assumptions, the fundamental class, the index class and the rho class of a spin Dirac operator on the regular part of a spin stratified pseudomanifold. More singular structures, such as singular foliations, are also treated. We employ groupoid techniques in a crucial way; however, an effort has been made in order to make this article accessible to readers with only a minimal knowledge of groupoids. Finally, whenever appropriate, a comparison between classical microlocal methods and groupoids methods has been provided.

Citations

• Introduction to Foliations and Lie Groupoids
• K-Theory for Operator Algebras
• Cyclic cohomology, the Novikov conjecture and hyperbolic groups
• Integrability of Lie brackets
• On the geometry of Riemannian manifolds with a Lie structure at infinity
• A cohomological formula for the Atiyah-Patodi-Singer index on manifolds with boundary

Cited by

• A Topological Approach to Indices of Geometric Operators on Manifolds with Fibered Boundaries