- Series
- Algebra Seminar
- Time
- Tuesday, April 17, 2012 - 2:05pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Jeremy Martin – University of Kansas – jmartin@math.ku.edu – http://www.math.ku.edu/~jmartin/
- Organizer
- Josephine Yu

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).