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arXiv · 1403.5518

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

Ayato Mitsuishi
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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.

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Ayato Mitsuishi. 2015-10-26. The coincidence of the current homology and the measure homology via a new topology on spaces of Lipschitz maps. https://arxiv.org/abs/1403.5518

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