# 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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