Cuts and Flows of Cell Complexes
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 constr...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2013-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/12794/pdf |
| _version_ | 1797270262734913536 |
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| author | Art M. Duval Caroline J. Klivans Jeremy L. Martin |
| author_facet | Art M. Duval Caroline J. Klivans Jeremy L. Martin |
| author_sort | Art M. Duval |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-25T02:01:29Z |
| format | Article |
| id | doaj.art-241eafebbf994cb88df1f5fa6c686019 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:01:29Z |
| publishDate | 2013-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-241eafebbf994cb88df1f5fa6c6860192024-03-07T14:52:35ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502013-01-01DMTCS Proceedings vol. AS,...Proceedings10.46298/dmtcs.1279412794Cuts and Flows of Cell ComplexesArt M. Duval0Caroline J. Klivans1Jeremy L. Martin2Department of Mathematical SciencesDepartment of MathematicsDepartment 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.https://dmtcs.episciences.org/12794/pdfcut latticeflow latticecritical groupspanning forestcell complex[info.info-dm]computer science [cs]/discrete mathematics [cs.dm] |
| spellingShingle | Art M. Duval Caroline J. Klivans Jeremy L. Martin Cuts and Flows of Cell Complexes Discrete Mathematics & Theoretical Computer Science cut lattice flow lattice critical group spanning forest cell complex [info.info-dm]computer science [cs]/discrete mathematics [cs.dm] |
| title | Cuts and Flows of Cell Complexes |
| title_full | Cuts and Flows of Cell Complexes |
| title_fullStr | Cuts and Flows of Cell Complexes |
| title_full_unstemmed | Cuts and Flows of Cell Complexes |
| title_short | Cuts and Flows of Cell Complexes |
| title_sort | cuts and flows of cell complexes |
| topic | cut lattice flow lattice critical group spanning forest cell complex [info.info-dm]computer science [cs]/discrete mathematics [cs.dm] |
| url | https://dmtcs.episciences.org/12794/pdf |
| work_keys_str_mv | AT artmduval cutsandflowsofcellcomplexes AT carolinejklivans cutsandflowsofcellcomplexes AT jeremylmartin cutsandflowsofcellcomplexes |