Maslov index, lagrangians, mapping class groups and TQFT

Abstract. Given a mapping class f of an oriented surface Σ and a lagrangian λ in the first homology of Σ, we define an integer nλ(f). We use nλ(f) (mod 4) to describe a universal central extension of the mapping class group of Σ as an index-four subgroup of the extension constructed from the Maslov index of triples of lagrangian subspaces in the homology of the surface. We give two descriptions of this subgroup. One is topological using surgery, the other is homological and builds on work of Turaev and work of Walker. Some applications to TQFT are discussed. They are based on the fact that our construction allows one to precisely describe how the phase factors that arise in the skein theory approach to TQFT-representations of the mapping class group depend on the choice of a lagrangian on the surface. Content