THE RESHETIKHIN-TURAEV REPRESENTATION OF THE MAPPING CLASS GROUP AT THE SIXTH ROOT OF UNITY

The quantum group construction of Reshetikhin and Turaev provides representations of the mapping class group, indexed by an integer parameter r. This paper presents computations of these representations when r=6, and analyzes their relationship to other topological invariants. It is shown that in genus 2, the representation splits into two summands. The first summand factors through the mapping class group action on the first homology of the surface with Z/3Z coefficients, while the second summand can be analyzed via its restriction to the subgroup of the mapping class group which is normally generated by the sixth power of a Dehn twist on a nonseparating curve. This analysis reveals a connection to the homology intersection pairing on the surface, and also yields information about the kernel and image of the representation. It is also shown that the representation yields a family of 2-dimensional nonabelian representations of the Torelli group.

This paper continues the program established by the author in [Wr] to relate the Reshetikhin-Turaev representations at specific roots of unity to classical invariants.