Discrete homology theory for metric spaces

Abstract

Abstract. 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.

Fundamental groups as limits of discrete fundamental groups

“…A homotopy between two paths γ, γ : [0, 1] → X must keep the endpoints fixed, while a free homotopy is allowed to move them. A closed path or loop is a path γ : [0, 1] → X with γ(0) = γ (1). We say that a loop is null-homotopic if it is homotopic to a constant path or, equivalently, if it is freely homotopic to a constant path (a free homotopy of loops is not allowed to break the loops into open paths).…”

“…Given two θ-paths Z 1 and Z 2 of the same length n, a free θ-grid homotopy between them is a θ-Lipschitz map H : If the endpoint of a θ-path coincides with the starting point of a second θ-path, the two θ-paths can be concatenated. Since the operation of concatenation is clearly compatible with θ-homotopies, we can make the following definition: [1]. Let x 0 be a base point in a metric space X.…”

“…The basic idea is similar to that of Example 5.2, but it is a little more technical. Instead of adding to the graph of sin(π/x) three squares, the space X is obtained adding a whole triangular prism built over the triangle with vertices (0, −1), (0, 1), (1,1). Moreover, we also add on the vertical plane passing through (0, −1), (1, −1) the subset of R 2 obtained by filling in the space contained between the graph of sin(π/x) and the x-axis as F (x 1 ) Figure 5.…”

“…On the other hand, the behaviour of π 1,θ (X, x 0 ) as θ tends to zero has not yet been investigated thoroughly. The following has been asked in [1,Section 7]: Question 1.1. Can the fundamental group π 1 (X, x 0 ) be recovered as some sort of limit of π 1,θ (X, x 0 ) as θ tends to 0?…”

Discrete cubical and path homologies of graphs

“…Later work by Babson, Barcelo, de Longueville, and Laubenbacher [1] connects this theory to classical homotopy theory of cubical sets and asks for a corresponding homology theory. The homology theory developed in [2] is an answer to that question. A more general but closely related homotopy theory for directed graphs was developed by Grigor'yan, Lin, Muranov, and Yau in [6], which also introduces a corresponding homology theory based on directed paths.…”