Critical groups of simplicial complexes

Details Critical groups of simplicial complexes Critical groups of simplicial complexes

2011-01-20

arxiv.org/abs/1101.3981v2

Mathematics

Combinatorics

14 pages, 2 figures; added Remark 4.7, relating to work of Molly Maxwell

Scientific paper

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.

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