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 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.
Additional Metadata
Keywords homology theory, discrete metric spaces
THEME Null option (theme 11)
Publisher Oxford U.P.
Journal Bulletin of the London Mathematical Society
Project Combining Machine Learning and Game-theoretic Approaches for Cluster Analysis
Grant This work was funded by the The Netherlands Organisation for Scientific Research (NWO); grant id nwo/612.001.352 - Combining Machine Learning and Game-theoretic Approaches for Cluster Analysis
Citation
Barcelo, H, Capraro, V, A. White, J, & Barcelo, H. (2014). Discrete homology theory for metric spaces. Bulletin of the London Mathematical Society, 46, 889–905.