Integral TQFT's, cut-numbers, and the mapping class groups
by
Thomas Kerler
The Ohio State University

The TQFT's of Reshetikhin and Turaev imply invariants of 3-manifolds as well as representations of the mapping class groups. Results by Murakami, Massbaum and Roberts for closed manifolds and Gilmer for manifolds with boundaries imply that the TQFT's can be made to live over the ring of cyclotomic integers Z[q], with q and r-th root of unity. We identify an explicit basis for the case r=5 and show that in this case we obtain a "half-projective TQFT" with non-invertible projective parameter h=(q-1) in Z[q]. A consequence of this is that the order of the invariant of a closed connected 3-manifold M in h yields an upper bound for the maximal number of surfaces that can be removed without disconnecting M, which is equal to the free rank of the fundamental group of M. 0-surgeries along almost boundary links and mapping tori provide large families of examples in which this estimate allows a precise determination of the cut numbers (joint with Pat Gilmer). For mapping tori we also use knowledge of the structure of the implied representations of the mapping class groups over Z[q] and Z/5. Time permitting we also discuss the latter in terms of expasions of matrices in h=q-1 and their relations to Torelli, boundary and further subgroups. An immediate consequence of this analysis is a new proof that the Ohtsuki-invariants (mod 5) are of finite type. Moreover we discuss implied formulae for the Casson invariant (mod 5) via representations of the symplectic groups and the Johnson homomorphisms.

Integral TQFT's, cut-numbers, and the mapping class groups

Date received: March 20, 2001