I am thinking about a problem as follows: Suppose a simply-connected 2-dimensional manifold has an $S^1$ boundary, is it homeomorphic to the open disk $D^2$? In fact, I would like to understand the general higher-dimensional case, i.e., how to decide the homeomorphism type of a (connected) manifold $M$ via its boundary submanifold $N$? Of course we need to put some condition on $M$ such like contractible or else, otherwise it could be arbitrary. In particular, when could we know that $M$ is the trivial fill-in manifold of $N$? For example, $M$ is the solid $k$-torus whereas $N=T^k=S^1\times\cdots\times S^1$.
I have no idea whether this is a research problem or just an exercise. I am sorry about my few knowledge on topology. If it is quite standard, could you please suggest me some reference books? Thanks a lot.