Homotopy lifting property of an e^ε-Lipschitz and co-Lipschitz map

2012/11/26 by Shicheng Xu, Xu, Shicheng
Mathematics · #53C21 #53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1211.5919

openalex publication_date 2012/11/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

Abstract

An e^ε-Lipschitz and co-Lipschitz map, as a metric analogue of an ε-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.

Citations

• On Almost Isometry Theorem in Alexandrov Spaces with Curvature Bounded Below
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Perelman's Stability Theorem

Related

• Lipschitz homotopy convergence of Alexandrov spaces
• Lipschitz homotopy convergence of Alexandrov spaces II
• Lipschitz homotopy and density of Lipschitz mappings in Sobolev spaces
• Topological Stability and Latschev-type Reconstruction Theorems for Spaces of Curvature Bounded Above
• Moderately Discontinuous Homotopy