Quantum Invariants and Representations of 3-Manifold Groups

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DMS-0207030Charles D. FrohmanThe PI will work on a number of projects in low dimensionaltopology and its applications. These projects include investigating various properties of the Turaev-Viroinvariants including the construction of universal polynomials relating the representation theory of the fundamental groups of manifolds to these invariants and the extension of these invariants away from the unit circle. In addition, the PI will consider, given a tetrahedral decomposition of a knot, how one may define a rigorouspath integral over the space of cross ratios of the tetrahedra to compute the Turaev-Viro invariant of an integral surgery on the knot. The PI will also investigate the structure of the Kauffman bracket skein module in terms of the geometry of character varieties,and, more generally, use quantum invariants in the study of Dehn surgery on knots. Finally, this award provides support for the PI's graduate students to assist him in this research. Topology is a kind of geometry where the congruence transformations do not preserve metric properties such as distance and angle. The objects the PI studies are given as the result of gluing together polyhedra along faces. To know when two such objects are different topologically one needs to make measurements that are unchanged by topological congruence transformations. An example of such a measurement is the Euler characteristic of a surface, which is the number of vertices minus thenumber of edges plus the number of faces. If two surfaces are topologically equivalent they have the same Euler characteristic. One of the most celebrated theorems of geometry is the Gauss-Bonnet theorem which relates the Euler characteristic of any surface to a quantity computed metrically. Quantum invariants of three manifolds are like Euler characteristic, but more delicate. They are constructed statistically from a probability space made up of "states" which come from the description of how polyhedra are glued together to form the object. The PI's work is about relating these invariants to metric quantities derived from the geometry of theunderlying object. To this end, the PI shows that the space of states can be replaced by spaces of geometric measurements made on the object. The goal of the project is to see how the geometry and topology of the object are determined by its combinatorial description in terms of polyhedra.