Stability of eϵ-Lipschitz and co-Lipschitz maps in Gromov-Hausdorff topology

A submetry is a metric analogue of a Riemannian submersion, and an eϵ-Lipschitz and co-Lipschitz map is a metric analogue of an ϵ-Riemannian submersion. The stability of submetries from Alexandrov spaces to Riemannian manifolds in the Gromov-Hausdorff topology can be viewed as a parametrized version of Perelman's stability theorem in Alexandrov geometry. In this paper, we will study the stability of eϵ-Lipschitz and co-Lipschitz maps. Our approach is based on controlled homotopy theory and semi-concave functions on Alexandrov spaces. As applications of our stability results, we generalize fiber bundle finiteness results on Riemannian submersions and partially generalize the stability of submetries.