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The following problem is a culmination of a few questions I've asked the last two months, and it's still giving me some issues. I think I know the right way to solve it, but I'm having trouble with the details; my idea can be formulated in terms of selection theorems/selectors or fiber bundles.

Let $X \subset \mathbb{R}^n$ be any subspace, and let $I = [0,1]$. By a proper isotopy of $X$ I mean a continuous function $F: X \times I \rightarrow \mathbb{R}^n$ such that for each $t \in I$, $f_t := F|_{X \times \lbrace t \rbrace}$ is an embedding and $f_0 = \text{id}_X$. By an ambient isotopy I mean an isotopy on all of $\mathbb{R}^n$.

If $F$ is a proper isotopy of a tamely embedded copy $X$ of $\mathbb{S}^{n-1}$ in $\mathbb{R}^n$, does $F$ extend to an ambient isotopy?

As noted in the answer here, this will be true as long as $F$ can be extended in some neighborhood of $X$. Let $In(X)$ denote the inside, i.e. bounded complementary domain, of $X$. By extending in the sphere from both sides of $X$ to get a neighborhood extension, it's equivalent to:

If $F$ is a proper isotopy of $X$, does $F$ extend to $In(X)$?

I'm mostly interested in $\mathbb{R}^2$. There it's already known, but the proof is very difficult. There was a follow-up paper where they broadened the result with different techniques; between the preprint, the Annals paper and the follow-up they gave three different arguments, but all hinged on analytically controlling crosscuts using geometric function theory to explicitly construct $F$.

Using results from the Kirby-Edwards paper linked in the previous MO thread, and a 'canonical' Alexander-Pontryagin Duality Theorem in the plane, you can prove the isotopy extension theorem for compact, connected subsets of $\mathbb{R}^2$ in a different way from the case for the circle (it's still somewhat complicated, but much less-so).

What I'd like to do is get the case for the circle using some selection theorem, esp. the Michael Selection Theorem (or even better, a selection theorem whose proof is actually reasonable). To do this, let $D$ be a large, closed ball around the trace of $X$ under $F$. By the Annulus Theorem, each region between $\partial(D)$ and $f_t(X) := X_t$ is a closed annulus, call it $A_t$.

For any $A_t$ there are many ways to partition it into crosscuts, so that each crosscut has one endpoint on each boundary component. By a crosscut, I mean an embedded copy of $I$. Let $\mathcal{C}_t$ denote the collection of such partitions on $A_t$, and let $\mathcal{C} = \cup \mathcal{C}_t$. Then what we want is a continuous selection of cross-cut decompositions, one for each $A_t$. To be precise, we should probably consider a family of parameterized cross-cut decompositions, so that each has a time parameterization (that will be our way of getting around $0$-regular convergence issues).

Alternatively, we can directly look for a continuous selection of crosscuts of the disc as follows. By interpolating with an ambient isotopy of affine shifts in the plane, we can assume that $0 \in In(f_t(X))$ for all $t$. Then we would let $\mathcal{C}$ be the collection of parametrized crosscut decompositions whose elements go through $0$.

Long story short, the problem for the selection method is:

How do you topologize (a suitable subset of) $\mathcal{C}$ to apply the Michael Selection Theorem?

This is equivalently a fiber bundle problem on $\mathbb{A}^n \times I$ (or the closed disc $\times I$) in the following sense. We have two cylinders, one the usual smooth cylinder, and the other one just some Jordan mess, as the boundaries. Can we warp it into the standard, smooth representative in a way that's slice?

The problem for the fiber bundle method is:

If $\mathbb{A}^n \times I \subset \mathbb{R}^{n+1}$ is a bundle over $I$ whose slices $\mathbb{A}^n_t$ are contained in hyperplaces orthogonal to the $(n+1)$-axis, is there an isotopy to the standard smooth (thickened) cylinder that's slice? In other words, $f_s(\mathbb{A}^n_t) \subset \mathbb{A}^n_t$ for all $s$ and $t$.

Thanks, appreciate any help at all!

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  • $\begingroup$ The fiber bundle version can also be asked for bundles over $\mathbb{S}^1$ where there is copious literature, just by 'reflecting, copying and attaching'. $\endgroup$ Apr 14 at 22:24
  • $\begingroup$ Visually it seems to me that the extension can even be PL off of $X$ (in dim 2/3). In the annulus formulation of the problem, you can swap the outer circle for a cube and approximate $F(X)$ with a countable sequence of rectilinear approximations from the outside. I assume you can bound the number of vertices at each level. Then all you need to do is make sure that any ray from the cube actually lands on $f_t(X)$, rather than converging to a non-trivial subset of it. Is there a theorem like this? Some sort of angular control for PL approximations of sphere isotopies? $\endgroup$ Apr 18 at 22:24
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Apparently exactly the same method (the Michael Selection Theorem method) was used here, in conjunction with some retract theory, to prove the theorem for graphs in the plane:

https://webusers.imj-prg.fr/~frederic.le-roux/RECHERCHE/TEXTES/0-LE%20ROUX-These-97.pdf

Especially chapter 3 of the appendix. But it's in frickin' French! The methods are definitely in the abstract-nonsense approach, I think I'll try to do it the dirty way with PL extensions. In particular, what are some conditions to extend a proper PL isotopy on a connected submanifold of $\mathbb{R}^n$ to the boundary? I can construct the approximation by rectilinear annuli and the isotopies between them, but I struggle to control the winding as you approach the boundary.

Is there a 'stretching, unwinding, and shrinking' method for extending PL isotopies?

I know of methods like this being used for the stable homeomorphism problem, they seem applicable here but I'm not versed in it.

Maybe the methods in that paper have application to the general, high-dimensional problem.

I think one powerful technique that might be possible (aside from assuming the inside constantly contains the origin) is to try to augment the original isotopy by another ambient isotopy that will make it so $In(f_s(J)) \subset In(f_t(J))$ for $s < t$. If you can do this then the trace will carve out an annulus and the continuous decomposition into trajectories will be trivial. Once you have the decomposition, you can define the isotopy along the trajectories.

I haven't worked out how to do it fully, but after a homeomorphism we can assume that $J$ is just the unit circle. Then if for each $\epsilon \in I$ you let $p(t) = (1+t) \cdot \max \lbrace \|f_s(x) - f_t(x)\|$ $|$ $x \in J, t-s \leq \epsilon \rbrace$, which is continuous and thus bounded on $I$. Then this defines an isotopy of radial expansions.

This does not work by itself, but if you let $G(x) = p(x) \cdot F(x)$ then it is an isotopy containing an increasing sequence of discs around the origin, especially starting with the unit disc (it grows faster than any point is traveling inward).

We need to further augment $p(x)$ by another isotopy which will force the sequence $In(g_t(J))$ to be increasing; I'm worried about any sort of thickening procedures creating some critical sets at some point along the way. I think it will be better to try to just make it convex directly - or at least star-convex relative to $0$, which is just as good.

To do that, I don't know the correct procedure. You have your maximal increasing discs, but then you have some 'tentacles' coming off them. If you visualize a very simple sort of tentacle, sort of concentric with the disc, then it's easy to understand how to adjust $p$ in polar coordinates to undo the 'rotative' part to get something star-convex wrt $0$. But if the tentacles are very complicated then I don't know how to define $p$, much less what sort of limiting argument you'd have to deal with.

Suppoe $K_t$ is the largest (closed) Euclidean disc containing $g_t(J)$ in its inside. I think the way to do it is to take some rays from $\infty$ to $K_t$ at each component, suck everything towards them radially, and then push back out in some way. Basically approximate a 2d Ricci flow. There will be some tree structure associated with the 'bad' parts of the disc, i.e. ones that span across multiple rays. You'll probably do this stretch-and-shrink at each tree level, one at a time.

In my head the limit processes that arise will work, but I don't even dare try to write the details. Somebody in differential equations or convex geometry should take over this method.

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