|Robert Penner (Aarhus Univ/USC/Caltech)
|Finite type invariants and the Ptolemy groupoid
|Tue, Mar 02, 2010, 13:15 - 14:45
|Seminar Room A
Recent joint work with Andersen, Bene, and Meilhan provides a new link between 2d geometry and 3d quantum topology. Specifically, we introduce a finite type invariant of suitable 3-manifolds which is universal for homology cylinders over a surface with boundary. This new invariant
depends upon a choice of fatgraph spine for the surface, and representations of its Ptolemy groupoid in the automorphism group of certain Jacobi diagrams evolve from this dependence. Applications include a TQFT-type
expression for the Le-Murakami-Ohtsuki invariant and a canonical cocycle, which originated in joint work with Morita, representing the first Johnson homomorphism.
The first 20-minute session will recall the Ptolemy groupoid and its utility for explicating representations of subgroups of the mapping class groups and then will formulate the basic problem of extending the LMO invariant in this context.