Cuts and flows of cell complexes

TL;DR

This paper extends the theory of cuts and flows from graphs to higher-dimensional cell complexes, providing explicit bases, topological interpretations, and relations to group invariants like the critical group.

Contribution

It develops a comprehensive framework for cuts and flows in cell complexes, including bases construction, topological interpretation, and connections to higher critical groups, generalizing graph theory results.

Findings

Constructed explicit bases for cut and flow spaces.

Linked discriminant groups to higher critical groups.

Provided bounds on complexity, girth, and connectivity.

Abstract

We study the vector spaces and integer lattices of cuts and flows associated with an arbitrary finite CW complex, and their relationships to group invariants including the critical group of a complex. Our results extend to higher dimension the theory of cuts and flows in graphs, most notably the work of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and give sufficient conditions for them to be integral bases of the cut and flow lattices. Second, we determine the precise relationships between the discriminant groups of the cut and flow lattices and the higher critical and cocritical groups with error terms corresponding to torsion (co)homology. As an application, we generalize a result of Kotani and Sunada to give bounds for the complexity, girth, and connectivity of a complex in terms of Hermite's constant.