Singular spaces, groupoids and metrics of positive scalar curvature

TL;DR

This paper develops a framework for analyzing spin Dirac operators on singular spaces using groupoid techniques, extending classical methods to stratified pseudomanifolds and singular foliations.

Contribution

It introduces new definitions for fundamental, index, and rho classes of Dirac operators on singular spaces using groupoids, bridging classical and modern approaches.

Findings

Defined fundamental, index, and rho classes for singular spaces

Extended analysis to stratified pseudomanifolds and foliations

Compared groupoid methods with classical microlocal techniques

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.