  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.
