Extending holomorphic motions and monodromy

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 Teichm\"uller spaces.