Discrete Homology Theory for Metric Spaces

Abstract

In this paper 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 verifies 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.

13 pages

References in corpus (2)

• Controlled coarse homology and isoperimetric inequalities
• Amenability, locally finite spaces, and bi-lipschitz embeddings

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• Homotopy, homology, and persistent homology using closure spaces
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• The Persistent Topology of Optimal Transport Based Metric Thickenings
• On the Čech cohomology of Morse boundaries