Lifting coarse homotopies

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