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Critical groups of simplicial complexes. (English) Zbl 1263.05124
Summary: 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.

MSC:
05E45 Combinatorial aspects of simplicial complexes
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C21 Flows in graphs
55U15 Chain complexes in algebraic topology
57M15 Relations of low-dimensional topology with graph theory
Software:
Macaulay2
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