Homotopy lifting property of an $e^{\epsilon }$-Lipschitz and co-Lipschitz map

Abstract

An $e^{\epsilon }$-Lipschitz and co-Lipschitz map, as a metric analogue of an $\epsilon$-Riemannian submersion, naturally arises from a sequence of Alexandrov spaces with curvature uniformly bounded below that converges to a space of only weak singularities. In this paper we prove its homotopy lifting property and its homotopy stability in Gromov-Hausdorff topology. Due to an overlook in the previous version, the part for bounding intrinsic distance of fibers will be talked about in the coming papers.

13 pages

References in corpus (2)

• Perelman's Stability Theorem
• On Almost Isometry Theorem in Alexandrov Spaces with Curvature Bounded Below