Lifting coarse homotopies

TL;DR

This paper advances coarse homotopy theory by establishing a lifting lemma for certain maps, enabling the analysis of coarse fundamental groups of quotients and metric cones over negatively curved manifolds.

Contribution

It proves a Coarse Lifting Lemma for a broad class of maps, including quotients by coarsely discontinuous actions, extending classical topological results to coarse geometry.

Findings

Established a Coarse Lifting Lemma for bornologous surjective maps.

Derived results on the coarse fundamental group of quotient spaces.

Computed the fundamental group of metric cones over negatively curved manifolds.

Abstract

Coarse geometry, and in particular coarse homotopy theory, has proven to be a powerful tool for approaching problems in geometric group theory and higher index theory. In this paper, we continue to develop theory in this area by proving a Coarse Lifting Lemma with respect to a certain class of bornologous surjective maps. This class is wide enough to include quotients by coarsely discontinuous group actions, which allows us to obtain results concerning the coarse fundamental group of quotients which are analogous to classical topological results for the fundamental group. As an application, we compute the fundamental group of metric cones over negatively curved compact Riemannian manifolds.