Critical Groups of Simplicial Complexes
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011)
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We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of abelian groups in terms of combinatorial Laplacian operators, generalizing the construction of the critical group of a graph. We show how to realize these critical groups explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe how the critical groups of a complex represent flow along its faces, and sketch another potential interpretation as analogues of Chow groups.
@article{DMTCS_2011_special_260_a22,
author = {Duval, Art M. and Klivans, Caroline J. and Martin, Jeremy L.},
title = {Critical {Groups} of {Simplicial} {Complexes}},
journal = {Discrete mathematics & theoretical computer science},
year = {2011},
volume = {DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011)},
doi = {10.46298/dmtcs.2909},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2909/}
}
TY - JOUR AU - Duval, Art M. AU - Klivans, Caroline J. AU - Martin, Jeremy L. TI - Critical Groups of Simplicial Complexes JO - Discrete mathematics & theoretical computer science PY - 2011 VL - DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) UR - http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2909/ DO - 10.46298/dmtcs.2909 LA - en ID - DMTCS_2011_special_260_a22 ER -
%0 Journal Article %A Duval, Art M. %A Klivans, Caroline J. %A Martin, Jeremy L. %T Critical Groups of Simplicial Complexes %J Discrete mathematics & theoretical computer science %D 2011 %V DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) %U http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2909/ %R 10.46298/dmtcs.2909 %G en %F DMTCS_2011_special_260_a22
Duval, Art M.; Klivans, Caroline J.; Martin, Jeremy L. Critical Groups of Simplicial Complexes. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), DMTCS Proceedings vol. AO, 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011) (2011). doi: 10.46298/dmtcs.2909
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