I have the following question:

For a given two-dimensional Riemann surface $C$,

Is there a way to classify all topologically distinct three-dimensional compact manifolds $M$ whose boundary is $C$, i.e., $\partial M =C$?

Is there always a three-dimensional compact manifold $M$ such that $\partial M =C$ and is contractible?

If there exists $M$ that satisfies condition 2, is it unique? If not unique, can the set of such manifolds be classified in a nice way?

Thank you in advance!