Building upon the work of Kalai and Adin, we extend the concept of a spanning tree from graphs to simplicial complexes, which are just higher-dimensional analogs of graphs. For any complex satisfying a mild technical condition, we show that its simplicial spanning trees can be enumerated using reduced Laplacian matrices, generalizing the Matrix-Tree Theorem.

We use these higher-dimensional spanning trees to extend the concept of a critical group of a graph (related to the sandpile model and the chip-firing game) to simplicial complexes. As in the graphical case, the critical group of a simplicial complex (if its codimension 1 skeleton has a suitably nice spanning tree) can be computed directly from the reduced Laplacian, and its order is given by a weighted count of the spanning trees.

This is joint work with Carly Klivans and Jeremy Martin.