There's an old problem in 4-manifold theory that, as far as I know, doesn't have a name associated with it and really deserves a name. 


Let $M$ be a smooth 4-manifold with boundary. Let $S$ be a smoothly embedded 2-dimensional sphere in $\partial M$. Assume $S$ does not bound a ball in $\partial M$, but $S$ is null-homotopic in $M$. Does $S$ bound a smooth 3-ball in $M$? Perhaps you need to replace $S$ by another non-trivial $S'$ in $\partial M$ before you can find a 3-ball in $M$ bounding it?


You could think of this as the co-dimension one analogue to Dehn's lemma for 4-manifolds. Usually when people talk about a Dehn lemma for 4-manifolds they're interested in the co-dimension 2 analogue. 


Does this problem / conjecture have a name? If not, do you have a good name for it? Do you know of anywhere in the literature where this issue is investigated? 


Off the top of my head the only vaguely related things I know about in the literature is a 1975 paper of Swarup's. 



A problem/conjecture related to 4-manifolds that deserves a name. What name does it deserve?
I think that the conjecture is wrong. The following leads to counterexamples in the topological category and probably also smoothly: Take a closed oriented 4-manifold N with infinite cyclic fundamental group and remove an open neighborhood of a generating circle. Then you get a 4-manifold M with boundary $S^1 \times S^2$ where only the generators can be represented by embedded 2-spheres. If S denotes $pt \times S^2$ then the following hold:


1. S is null homotopic in M via Poincare duality and the exact sequence of the pair.
2. S bounds an embedded 3-ball in M if and only if N can be decomposed into $S^1 \times S^3$, connected sum with a simply connected manifold.


Hambleton and I found topological 4-manifolds N for which the intersection form is not extended from the integers, in particular 2 doesn't hold. With Friedl and Melvin we showed later that our examples don't have a smooth structure. But now I remember a discussion with Fintushel and Stern who mentioned that they constructed a smooth counterexample to 2 (and hence to the conjecture).


