Art M. Duval ; Caroline J. Klivans ; Jeremy L. Martin - Cuts and Flows of Cell Complexes

dmtcs:12794 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.12794
Cuts and Flows of Cell ComplexesArticle

Authors: Art M. Duval 1; Caroline J. Klivans 2,3; Jeremy L. Martin 4

  • 1 Department of Mathematical Sciences
  • 2 Department of Computer Science
  • 3 Department of Mathematics
  • 4 Department of Mathematics [Kansas]

We study the vector spaces and integer lattices of cuts and flows of an arbitrary finite CW complex, and their relationships to its critical group and related invariants. Our results extend the theory of cuts and flows in graphs, in particular the work of Bacher, de la Harpe and Nagnibeda. We construct explicit bases for the cut and flow spaces, interpret their coefficients topologically, and describe 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; these are expressed as short exact sequences 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.


Volume: DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013)
Section: Proceedings
Published on: January 1, 2013
Imported on: November 21, 2016
Keywords: cut lattice,flow lattice,critical group,spanning forest,cell complex,[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]

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