On the Witten-reshetikhin-turaev Representations of Mapping Class Groups

We consider a central extension of the mapping class group of a surface with a collection of framed colored points. The Witten-ReshetikhinTuraev TQFTs associated to SU(2) and SO(3) induce linear representations of this group. We show that the denominators of matrices which describe these representations over a cyclotomic field can be restricted in many cases. In this way, we give a proof of the known result that if the surface is a torus with no colored points, the representations have finite image. Recall that an object in a cobordism category of dimension 2+1 is a closed oriented surface Σ, perhaps with some specified further structure. A morphism M from Σ to Σ′ is (loosely speaking) a compact oriented 3-manifold perhaps with some specified further structure, called a cobordism, whose boundary is the disjoint union of −Σ and Σ′. A morphism M ′ from Σ′ to Σ′′ is composed with a morphism from Σ to Σ′ by gluing along Σ′, inducing any required extra structure from the structures on M and M ′. Also, the extra structure on a 3-manifold must induce the extra structure on the boundary. A TQFT in dimension 2+1 is then a functor from such a cobordism category to the category of modules over some ring R. There are further axioms that are generally required [A], [BHMV], [Q]. One usually denotes the module associated to Σ by V (Σ), and denotes the homomorphism associated to cobordism M by Z(M). A TQFT yields a representation of (an extension) of the mapping class group of a surface. An extension is needed if there is some choice in the extra data which may be placed on a mapping cylinder. We will study a version of the Witten-Reshetikhin-Turaev TQFTs [W], [RT] associated to SU(2) and SO(3) constructed by [BHMV]. In particular, we will use the notation where Vp for p = 2r is a TQFT associated to SU(2). Also, Vp for p odd is a TQFT associated to SO(3). We will assume that p ≥ 3. We will use a variant of the [BHMV] approach obtained by adapting an idea of Walker [Wa]. We replace, in the definition of the cobordism category, a p1-structure on a surface Σ with a Lagrangian subspace of H1(Σ,Z), and a p1-structure on a 3-manifold M by an integer and a Lagrangian subspace for the boundary of M. If one does this, one may work over the ring rp = Z[Ap, 1 p , up], where Ap is a primitive 2pth root of unity, Received by the editors June 23, 1997 and, in revised form, November 5, 1997. 1991 Mathematics Subject Classification. Primary 57M99.