The coincidence of the current homology and the measure homology via a new topology on spaces of Lipschitz maps

2014/03/21 by Ayato Mitsuishi, Mitsuishi, Ayato
Computer Science · Mathematics · #Geometric Analysis and Curvature Flows #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.AT #math.GN #math.MG

paper · pdf · doi:10.48550/arxiv.1403.5518

arxiv created 2015/10/26 · arxiv updated 2015/10/27

Abstract

We consider the category of all locally Lipschitz contractible metric spaces and all locally Lipschitz maps, which is a wide class of metric spaces, including all finite dimensional Alexandrov spaces and all CAT spaces. We also consider the chain complex of normal currents with compact support in a metric space in the sense of Ambrosio and Kirchheim. In the present paper, its homology is proved to be a homotopy invariant on the category. To prove this result, we define a new topology on a space of Lipschitz maps between arbitrary metric spaces. This topology is proved to coincide with the usual C¹-topology on the space of C¹-maps between compact Riemannian manifolds.

Related

• The coincidence of the homologies of integral currents and of integral singular chains, via cosheaves
• Singular (Lipschitz) homology and homology of integral currents
• Homology by metric currents
• Lipschitz homotopy convergence of Alexandrov spaces
• Flat forms, bi-Lipschitz parametrizations, and smoothability of manifolds