# The adiabatic groupoid and the Higson-Roe exact sequence

## TL;DR

This paper establishes an isomorphism between the Higson-Roe analytic surgery sequence and the adiabatic deformation sequence of a Lie groupoid, extending to stratified manifolds and relating positive scalar curvature invariants.

## Contribution

It proves the isomorphism between the Higson-Roe sequence and the adiabatic deformation sequence for certain manifolds, extending to stratified cases and connecting scalar curvature invariants.

## Key findings

- Isomorphism between Higson-Roe and adiabatic deformation sequences
- Extension to smoothly stratified manifolds
- Identification of $
ho$-classes for positive scalar curvature

## Abstract

Let $\widetilde{X}$ be a smooth Riemannian manifold equipped with a proper, free, isometric and cocompact action of a discrete group $\Gamma$. In this paper we prove that the analytic surgery exact sequence of Higson-Roe for $\widetilde{X}$ is isomorphic to the exact sequence associated to the adiabatic deformation of the Lie groupoid $\widetilde{X}\times_\Gamma\widetilde{X}$. We then generalize this result to the context of smoothly stratified manifolds. Finally, we show, by means of the aforementioned isomorphism, that the $\varrho$-classes associated to a metric with positive scalar curvature defined by Piazza and Schick corresponds to the $\varrho$-classes defined by the author of this paper.