On the homology of locally finite graphs

TL;DR

This paper establishes a connection between the topological cycle space of locally finite graphs and their singular homology, introducing a new homology theory for non-compact spaces with ends that generalizes graph cycles.

Contribution

It demonstrates that the topological cycle space is a canonical quotient of the first singular homology group and introduces a novel singular-type homology applicable in any dimension.

Findings

Topological cycle space is a quotient of the first singular homology group.

Characterization of graphs where the cycle space and homology coincide.

A new homology theory for non-compact spaces with ends, capturing graph cycles in dimension 1.

Abstract

We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension~1 captures precisely the topological cycle space of graphs but works in any dimension.