The coincidence of the homologies of integral currents and of integral singular chains, via cosheaves

2013/03/31 by Ayato Mitsuishi, Mitsuishi, Ayato
Mathematics · #Algebraic Topology (math.AT) #FOS: Mathematics #Metric Geometry (math.MG) #math.AT #math.MG

paper · pdf · doi:10.48550/arxiv.1304.0152

Results and proofs are refined to be more readable. Along with those, the title is changed

arxiv created 2015/10/21 · arxiv updated 2015/10/22

Abstract

We consider the notion of metric spaces being locally Lipschitz contractible introduced by Yamaguchi, and a category of metric spaces satisfying this condition. Many objects in metric geometry including CAT-spaces and Alexandrov spaces, belong to this category. We consider the homology of integral currents with compact support in a metric space, introduced by Ambrosio and Kirchheim, and prove that it and the usual integral singular homology are isomorphic on the category. The proof of it is based on the theory of cosheaves. A method to compare the homologies associated to cosheaves is also proved in this paper.

Related

• The coincidence of the current homology and the measure homology via a new topology on spaces of Lipschitz maps
• Singular (Lipschitz) homology and homology of integral currents
• Homology by metric currents
• The pointed flat compactness theorem for locally integral currents
• Decomposition of integral metric currents