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

Research paper by Ayato Mitsuishi

Indexed on: 26 Oct '15Published on: 26 Oct '15Published in: Mathematics - Algebraic Topology

#### 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 \$C^1\$-topology on the space of \$C^1\$-maps between compact Riemannian manifolds.