The Reshetikhin-Turaev representation of the mapping class group.

The Reshetikhin-Turaev three-manifold invariant $\tau\sb{r}$ is obtained by using the representations of a modular Hopf algebra and a surgery description of the manifold. Closely associated to it is a representation of the mapping class group of an orientable surface. This representation is the subject of this paper. The main result of this paper is that when r = 4, the restriction of the representation to the Torelli group is equal to the sum of the Birman-Craggs homomorphisms. This result is proved by constructing an explicit correspondence between the basis vectors of the space on which the representation acts, and the $\doubz/2\doubz$-quadratic forms on the first homology of the surface. This result corresponds to the fact, shown by Kirby and Melvin, that the three-manifold invariant when r = 4 is related to spin structures on the associated four-manifold. The proof is carried out by computing the value of the representation on one generator of the Torelli group, and the conjugation action of the full mapping class group on the image of the Torelli group.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/103159/1/9303840.pdfDescription of 9303840.pdf : Restricted to UM users only