Critical groups of simplicial complexes

TL;DR

This paper extends the concept of critical groups from graphs to simplicial complexes, defining new algebraic invariants that relate to combinatorial Laplacians and spanning trees.

Contribution

It introduces a generalized theory of critical groups for simplicial complexes, including explicit constructions and properties, expanding graph theory concepts to higher-dimensional structures.

Findings

Critical groups are finite and their order relates to weighted spanning trees.

Explicit realization of critical groups as cokernels of reduced Laplacians.

Potential interpretation as analogues of Chow groups.

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.