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Critical Groups of Simplicial Complexes

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Abstract

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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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Texas at El Paso, El Paso, Texas, 79968-0514, USA

    Art M. Duval

  2. Departments of Mathematics and Computer Science, The University of Chicago, Chicago, IL, 60637, USA

    Caroline J. Klivans

  3. Department of Mathematics, University of Kansas, Lawrence, Kansas, 66045-7594, USA

    Jeremy L. Martin

Authors
  1. Art M. Duval
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  2. Caroline J. Klivans
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  3. Jeremy L. Martin
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Corresponding author

Correspondence to Jeremy L. Martin.

Additional information

The third author was supported in part by National Security Agency Young Investigators Grant number H98230-08-1-0073-P0001.

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Duval, A.M., Klivans, C.J. & Martin, J.L. Critical Groups of Simplicial Complexes. Ann. Comb. 17, 53–70 (2013). https://doi.org/10.1007/s00026-012-0168-z

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  • DOI: https://doi.org/10.1007/s00026-012-0168-z

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