Title: Extending holomorphic motions and monodromy,
Abstract: Let $E$ be a closed set in the Riemann sphere $\widehat{\mathbb{C}}$. We
consider a holomorphic motion $\phi$ of $E$ over a complex manifold $M$, that
is, a holomorphic family of injections on $E$ parametrized by $M$. It is known
that if $M$ is the unit disk $\Delta$ in the complex plane, then any
holomorphic motion of $E$ over $\Delta$ can be extended to a holomorphic motion
of the Riemann sphere over $\Delta$. In this paper, we consider conditions
under which a holomorphic motion of $E$ over a non-simply connected Riemann
surface $X$ can be extended to a holomorphic motion of $\widehat{\mathbb{C}}$
over $X$. Our main result shows that a topological condition, the triviality of
the monodromy, gives a necessary and sufficient condition for a holomorphic
motion of $E$ over $X$ to be extended to a holomorphic motion of
$\widehat{\mathbb{C}}$ over $X$. We give topological and geometric conditions
for a holomorphic motion over a Riemann surface to be extended. We also apply
our result to a lifting problem for holomorphic maps to Teichmüller spaces.