# Smooth Schoenflies theorem for compact $3$-manifolds

Let $$M^3$$ be a compact $$3$$-manifold with $$\partial M=N$$ a connected surface. Suppose one has a smooth embedding of $$N$$ into the interior of $$M$$ and $$N$$ bounds a domain $$D$$ in $$M$$. Can we show that $$D$$ is homeomorphic to $$M$$?

• This is certainly not true. One class of examples is given by the complements of satellite knots: en.wikipedia.org/wiki/Satellite_knot. These contain tori that are not parallel to the boundary, and indeed these tori bound a "deeper" knot complement.
– HJRW
Jun 10, 2021 at 10:23
• If you drop the dimensions down to $2$, it is also not true. There is however a way to phrase it for curves. Jun 10, 2021 at 15:23
• This is true only if M is the 3-ball. Jun 11, 2021 at 15:21
• @BrunoMartelli Why not for other handlebodies? Jun 20, 2021 at 23:57
• You can embed a surface of genus >= 1 inside a ball in a strange way, so that it does not bound a handlebody in it. Jun 21, 2021 at 14:02

Here is a concrete example. Suppose that $$K \subset S^3$$ is the trefoil knot in the three-sphere. It is an old result (certainly known to Alexander) that $$K$$ is not isotopic to the unknot. Let $$N(K)$$ be a small tubular neighbourhood of $$K$$, taken in $$S^3$$. So $$N(K)$$ is a solid torus. Let $$n(K)$$ be the interior of $$N(K)$$. We define $$M = S^3 - n(K)$$. This is a (compact) knot exterior. By the "old result" $$M$$ is not homeomorphic to a solid torus. Note that $$T = \partial M = \partial N(K)$$ is a two-torus.
Now choose an embedded loop $$\alpha$$ in $$M$$. Then $$N(\alpha)$$, a small tubular neighbourhood of $$\alpha$$, is another solid torus. Again $$T' = \partial N(\alpha)$$ is a two-torus. However, $$N(\alpha)$$ is not homeomorphic to $$M$$.