Discrete homology theory for metric spaces

By H. Barcelo, Valerio Capraro, J. White. 2013

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We define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an n ‐dimensional cube to a fixed metric space. We prove that the resulting homology theory satisfies a discrete analogue of the Eilenberg–Steenrod axioms, and prove a discrete analogue of the Mayer–Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a ‘fine enough’ rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non‐trivial coarse invariant of a metric space.

Subfields: Computational Theory and Mathematics, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bf181038e1e0868e946f251f90522e57518ffec3

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In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Homologies of digraphs and Künneth formulas (2017, background): “More recently there have been a number of attempts to define singular homologies of graphs [2], [17].”