Question

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.

Answer 1

In dimension 3, you have the sphere theorem, the torus theorem, the annulus theorem, and the disk theorem (which is the loop theorem and Dehn's lemma put together).

So, if you didn't require the sphere to be embedded, and the problem already had an affirmative answer, I'd call it the Ball Theorem.