The adiabatic groupoid and the Higson–Roe exact sequence

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 a positive scalar curvature defined by Piazza and Schick (2014) correspond to the \varrho -classes defined by Zenobi (2019).