Semi-coarse Spaces, Homotopy and Homology

TL;DR

This paper introduces semi-coarse spaces, extending coarse geometry to compact metric spaces, and develops their homotopy and homology theories, linking graph structures to Vietoris-Rips homology.

Contribution

It defines semi-coarse spaces, constructs their homology, and establishes a homotopy invariance theorem connecting graph-induced semi-coarse homology with Vietoris-Rips homology.

Findings

Semi-coarse spaces generalize coarse spaces for compact metric spaces.

Homology groups are invariant under semi-coarse homotopy.

Graph-induced semi-coarse homology is isomorphic to Vietoris-Rips homology.

Abstract

We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial `coarse-like' structures to compact metric spaces, something which is impossible in coarse geometry. We first study homotopy in this context, and we then construct homology groups which are invariant under semi-coarse homotopy equivalence. We further show that any undirected graph $G=(V,E)$ induces a semi-coarse structure on its set of vertices $V_G$, and that the respective semi-coarse homology is isomorphic to the Vietoris-Rips homology. This, in turn, leads to a homotopy invariance theorem for the Vietoris-Rips homology of undirected graphs.