Questions on 3-manifolds with a given boundary 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!
 A: (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)
A: *

*When you say "Riemann surface", do you mean "topological surface"? Does the Riemann surface structure have any significance?


I assume below that you mean "two-manifold"


*

*Well, any three manifold contains a handlebody of your favorite genus, so this question is at least as hard as classifying three-manifolds (which is possible, by Thurston-Perelman).

*No, by the half-lives half-dies theorem.
3.Vacuous, because of 2.
A: 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.
