In genus at least four, the mapping class group of an orientable surface has a universal central extension by Z. The aim of this talk is to construct this extension explicitly as an index four subgroup of the extension coming from the Maslov index of triples of lagrangian subspaces in the first homology of the surface. The key ingredient of our construction is the definition of an integer n_\lambda(f) for every pair consisting of a mapping class f and a lagrangian subspace \lambda. This construction was motivated by the representations of mapping class groups arising in integral TQFT. (Joint work with P. M. Gilmer.).