Discrete Homology Theory for Metric Spaces

TL;DR

This paper introduces a novel discrete homology theory for metric spaces using Lipschitz maps, establishing foundational properties, connections to homotopy, and applications to manifold fundamental groups and coarse invariants.

Contribution

It develops a new discrete homology framework for metric spaces, proving axiomatic properties, relating it to homotopy, and applying it to manifold groups and coarse geometry.

Findings

Homology satisfies discrete Eilenberg-Steenrod axioms.

Discrete homology relates to discrete homotopy via a Hurewicz theorem.

Fundamental groups of manifolds can be captured by discrete homology.

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.