Let $F$ be a compact oriented surface and $\rho:\pi_1(F)\rightarrow SL_2\mathbb{C}$ be a representation. Does there exist a compact oriented three-manifold $M$ with $\partial M=F$ and a homomorphism $\tilde{\rho}:\pi_1(M)\rightarrow SL_2\mathbb{C}$ so that the restriction of $\tilde{\rho}$ to $\pi_1(F)$ is equal to $\rho$?

If not is there an obstruction that allows you to identify the representations that do extend?

For instance let $BSL_2\mathbb{C}^\delta$ denote the classifying space of $SL_2\mathbb{C}$ as a discrete group. Corresponding to $\rho:\pi_1(F)\rightarrow SL_2\mathbb{C}$ is a continuous map $f:F\rightarrow BSL_2\mathbb{C}^\delta$. If $\rho$ extends over a three-manifold $M$ then the homology class represented by $f_\#[F]$ is zero. Is there a computable way to detect this?