# Fox re-imbedding theorem in dimension four

Fox re-imbedding theorem states the following:

A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of handlebodies.

The proof uses some standard three-dimensional arguments, like cutting along essential discs (which exist either in $M$ or in its complement, since the boundary components are compressible) and proceeding by some induction. Note that the boundary of $M$ may be disconnected.

I am wondering whether the same result holds in higher dimension:

Let $M$ be a compact 4-manifold with boundary that embeds (smoothly) in $S^4$. Does it (smoothly) embed in $S^4$ so that its complement is a union of 2-handlebodies?

A 2-handlebody here is a 4-manifold that decomposes only in 0-, 1-, and 2-handles.

Remark: We need 2-handles here, because without them we would only get 4-manifolds with very special boundary ($S^3$ or a connected sum of $S^2\times S^1$'s). Every closed connected 3-manifold is the boundary of a 2-handlebody.

The question is whether one can avoid 3-handles - this is a standard type of question in dimension four and it is typically difficult; however, maybe in this special case something is known. There is also a simpler version:

Let $M$ be a compact 4-manifold with boundary that embeds (smoothly) in a homotopy sphere. Does it (smoothly) embed in some other homotopy sphere so that its complement is a union of 2-handlebodies?

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I just stumbled upon your question. My brain isn't quite tuned to it yet, but the Lickorish paper "Knotted contractible 4-manifolds in $S^4$" seems potentially relevant. This is in Pac. J. Math, 208 no. 2 (2003). – Ryan Budney Jan 23 '15 at 7:30
I had asked a similar question one of my papers, on embedding 3-manifolds in $S^4$. If a $3$-manifold admits a smooth embedding in $S^4$ is it possible to re-embed it so that fundamental groups of the two exterior components have solvable word problem? – Ryan Budney Jan 23 '15 at 17:52
More generally, which re-imbedding techniques do we know? If I have a 3-manifold in S^4, how can I modify its embedding so that it is not isotopic to the previous one? – Bruno Martelli Jan 23 '15 at 19:10
Rubinstein Pac. J. Math 86 (1980) has a result (due to Aitchinson) that an embedded $S^1 \times S^2$ in $S^4$ has a complementary component homeomorphic to $S^2 \times D^2$, so the other side is a $2$-knot complement. – Ryan Budney Feb 3 '15 at 23:33