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 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 $X$.

I’m mostly interested in the case for the plane. In fact, for the plane it’s already known to be true, but the proof is very difficult. There was a follow-up paper where they defined a notion of length for plane curves that extended to unrectifiable curves, and which behaved continuously with respect to uniform convergence of compact sets (so basically, they didn’t require 0-regular convergence); 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. My idea is also to use cross-cuts, but not to construct an explicit isotopy.

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 (though it’s also fairly complicated, and is ill-suited to the generalizations they obtained in the follow-up).

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).

This is equivalently a fiber bundle problem on $mathbb{A}^n times I$ 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*?

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

How do you topologize $mathcal{C}$ to apply the Michael Selection Theorem?

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!