## Abstract.

Given a mapping class *f* of an oriented
surface and a
lagrangian λ in the first homology of , we define an
integer
. We use
to describe
a
universal central extension of the
mapping class group of 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.

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