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 $H_1$. This homology seems to be particularly apt for studying spaces with infinitely generated $H_1$, e.g. infinite graphs or fractals.

Advances in Mathematics (2011)

References in corpus (3)

• Locally finite graphs with ends: a topological approach
• Uniqueness of electrical currents in a network of finite total resistance
• Coverings and fundamental groups: a new approach

Cited by in corpus (4)

• Gas dynamics in Massive Dense Cores in Cygnus-X
• On prisms, Möbius ladders and the cycle space of dense graphs
• On the coincidence of zeroth Milnor-Thurston homology with singular homology
• On spanning tree packings of highly edge connected graphs