Question: I have the following question:

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

1. Is there a way to classify all topologically distinct
three-dimensional compact manifolds $M$ whose boundary is $C$,
i.e., $\partial M =C$?
2. Is there always a three-dimensional compact manifold
$M$ such that $\partial M =C$ and is contractible?
3. 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!

Answer: 
(1) Consider the case where C is a sphere. For any such manifold, we can just add a 3-dimensional ball to get a closed compact 3-fold. So in this case it's just the classification of closed compact 3-folds.

For the torus, it's slightly more compact. One can again add a donut, so the problem includes the classification of closed compact 3-folds. But suppose this 3-fold turns out to be a sphere. We still don't know how the donut embeds - in particular, it could surround any knot. So it also includes the classification of knots.

(3) This is not entirely vacuous because it is a reasonable question to ask if C is a sphere. In this case, M, a closed ball, is unique up to homeomorphism, by the Poincare conjecture. (Glue a closed ball to it, get a homotopy 3-sphere, which is therefore homeomorphic to a 3-sphere)
For (3), here is a nice counterexample if we ignore contractibility (taken from an article from *The American Mathematical Monthly*):

<http://www.jstor.org/stable/pdfplus/2695643.pdf>

Hopefully someone can embed the image (Figure 13). It is two topologically distinct manifolds with the torus as a boundary... one being a solid [knot] torus, and the other being a 'cylinder' with a knotted inner-hole.