Cycle decompositions: From graphs to continua

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 H1. This homology seems to be particularly apt for studying spaces with infinitely generated H1, e.g. infinite graphs or fractals.