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I at first thought this should be an easy question, but then realized it might actually not even be known.

Let us work in the smooth category, though I am also wondering about PL and TOP (locally flat). Let $D_1, D_2$ be codimension-1 disks in an n-sphere. Suppose $\partial D_1 =\partial D_2$. Is there an ambient isotopy which makes $D_1 = D_2$, relative to their shared boundary?

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  • $\begingroup$ Is it not Schoenflies conjecture? $\endgroup$
    – Anubhav Mukherjee
    Commented Apr 8 at 1:48
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    $\begingroup$ If you replace the 4-sphere with a simply-connected 4 manifold, then one of my earlier papers I showed the existence of two 3-balls which are topologically isotopic rel boundary but not smoothly. $\endgroup$
    – Anubhav Mukherjee
    Commented Apr 30 at 15:35

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This is a little different than the Schoenflies problem.

You can rephrase your question to be about the space of embeddings

$$D^{n-1} \to S^1 \times D^{n-1}$$

that agree with the standard embedding $\{1\} \times D^{n-1} \to S^1 \times D^{n-1}$ on the boundary.

i.e. for your embedding drill out a neighbourhood of the boundary, and you get (essentially) one of these.

The Schoenflies problem asks if when you lift an embedding

$$D^{n-1} \to S^1 \times D^{n-1}$$

to the universal cover

$$D^{n-1} \to \mathbb R \times D^{n-1}$$

then is it isotopic to an embedding with image $\{0\} \times D^{n-1}$, i.e. the linear embedding.

What we know is the answer to your question is yes in dimensions $2$ (*) and $3$ but false for $n=4$ and $n \geq 6$. The $n=4$ case was first proven by David Gabai and myself. Tadayuki Watanabe has come up with an argument for the $n=4$ case, as well. The $n \geq 6$ case was done much earlier by Hatcher and Wagoner. The $n=5$ case is likely false as well, and it should be the topic of a paper to be put on the arXiv soon.

(*) Technically to make my setup equivalent to yours, in the $n=2$ case you need to demand the embedding $D^1 \to S^1 \times D^1$ does not link/wind around the $S^1$-factor.

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  • $\begingroup$ Dear Ryan, can you tell me a little more about your comment " The n=5 case is likely false...". I am curious to know the idea. $\endgroup$
    – Anubhav Mukherjee
    Commented Apr 11 at 14:47
  • $\begingroup$ @AnubhavMukherjee: we have an argument that the Hatcher-Wagoner $\theta$ invariant is computable via the "scanning" type maps, as in my two preprints with Gabai. So that deals with the $n \geq 6$ case, but the underlying homotopy theory is somewhat different than that $n=4$ case. The $n=5$ case we show there is a similar non-trivial scanning invariant, but again the underlying algebraic topology is different than the $n=4$ case and the $n \geq 6$ case. $\endgroup$
    – Ryan Budney
    Commented Apr 11 at 15:07
  • $\begingroup$ If you replace the 4-sphere with a simply-connected 4 manifold, then one of my earlier papers I showed the existence of two 3-balls which are topologically isotopic rel boundary but not smoothly. $\endgroup$
    – Anubhav Mukherjee
    Commented Apr 30 at 15:34

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