Cellular Cuts, Flows, Critical Groups, and Cocritical Groups

Algebra Seminar
Tuesday, April 17, 2012 - 14:05
1 hour (actually 50 minutes)
Skiles 006
University of Kansas
The critical group of a graph G is an abelian group K(G) whose order is the number of spanning forests of G.  As shown by Bacher, de la Harpe and Nagnibeda, the group K(G) has several equivalent presentations in terms of the lattices of integer cuts and flows on G. The motivation for this talk is to generalize this theory from graphs to CW-complexes, building on our earlier work on cellular spanning forests. A feature of the higher-dimensional case is the breaking of symmetry between cuts and flows.  Accordingly, we introduce and study two invariants of X: the critical group K(X) and the cocritical group K^*(X), As in the graph case, these are defined in terms of combinatorial Laplacian operators, but they are no longer isomorphic; rather, the relationship between them is expressed in terms of short exact sequences involving torsion homology. In the special case that X is a graph, torsion vanishes and all group invariants are isomorphic, recovering the theorem of Bacher, de la Harpe and Nagnibeda.  This is joint work with Art Duval (University of Texas, El Paso) and Caroline Klivans (Brown University).