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!



Questions on 3-manifolds with a given boundary
1. 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"


1. 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).
2. No, by the half-lives half-dies theorem.


3.Vacuous, because of 2.


