Cycle decompositions: from graphs to continua

TL;DR

This paper extends a key graph theory fact about cycle decompositions to general continua by introducing a new homology group, enabling analysis of complex spaces like fractals and infinite graphs.

Contribution

It introduces a novel homology group for continua that generalizes cycle decompositions from graphs to topological spaces.

Findings

Generalizes cycle decomposition to continua using topological circles

Defines a new homology group suitable for complex spaces

Applicable to spaces with infinitely generated $H_1$

Abstract

We generalise a fundamental graph-theoretical fact, stating that every element of the cycle space of a graph is a sum of edge-disjoint cycles, to arbitrary continua. To achieve this we replace graph cycles by topological circles, and replace the cycle space of a graph by a new homology group for continua which is a quotient of the first singular homology group $H_1$. This homology seems to be particularly apt for studying spaces with infinitely generated $H_1$, e.g. infinite graphs or fractals.