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In the smooth setting, Whitney's approximation theorem says the following: If $M,N$ are smooth manifolds and $f,g:M\to N$ are smooth functions that are continuously homotopic (ie there is a continuous homotopy $H:M\times [0,1]\to N$ between them) then they are also smoothly homotopic (ie there exists a smooth homotopy $\tilde{H}$ between them).

Does something similar hold for Lipschitz functions between Lipschitz manifolds? More precisely, if $M,N$ are two Lipschitz manifolds and $f,g:M\to N$ are Lipschitz functions that are continuously homotopic, is it true that they are also Lipschitz homotopic (ie there exists a Lipschitz homotopy between them)?

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    $\begingroup$ see mathoverflow.net/questions/295213/lipschitz-homotopy-groups. Apparently, Heisenberg group, which is contractible, has some nontrivial Lipschitz homotopy groups. $\endgroup$ Feb 9, 2022 at 20:44
  • $\begingroup$ Thanks. But the Heisenberg group is a smooth manifold, right? So how does this reconcile with the comment (in the question you linked) that says that for smooth manifolds smooth and Lipschitz homotopy groups coincide? $\endgroup$
    – Titti
    Feb 9, 2022 at 21:02
  • $\begingroup$ Yes, Heisenberg group is a Lie group, hence a smooth manifold. Lipschitz homotopy groups depend on the metric, and in this case it is the Carnot-Carateodory metric, see arxiv.org/pdf/1210.6943.pdf. There could be Lipschitz spheres that are smoothly fillable but not Lispschitz fillable. $\endgroup$ Feb 9, 2022 at 21:10
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    $\begingroup$ I see. But I am afraid that my question remains open then, since in the example of the Heisenberg group the Lipschitz structure is not the one induced by the Lipschitz manifold structure... $\endgroup$
    – Titti
    Feb 9, 2022 at 21:18
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    $\begingroup$ The identity and the negation are two Lipschitz functions $\mathbb R\to\mathbb R$ that are homotopic, but not Lipschitz homotopic. . . . The Lipschitz condition has both small-scale and large-scale restrictions. Most people only really want one and should specify which. If you restrict to compact manifolds, it's just a small-scale condition, whereas this example is a coarse phenomenon. $\endgroup$ Feb 10, 2022 at 16:17

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For compact Lipschitz manifolds this follows from the main result of the paper by Liu, Luofei, Yu, Hanfu, Liu, Ye "Converting uniform homotopies into Lipschitz homotopies via moduli of continuity." Topology Appl. 285 (2020)

But the above paper is surely an overkill. It gives quantitative bounds on the homotopy and qualitative result should be known much earlier.

I expect it's known that for a compact Lipschitz manifold $N$, for some Lipschitz embedding of $N$ into some $\mathbb R^n$ a small neighbourhood of $N$ Lipschitz retracts onto $N$. This would easily imply the result.

It's likely even true that $N$ is a Lipschitz ANR.

It's also clear that there is an appropriate version of this for noncompact manifolds too.

Edit: I found a reference to the fact that any Lipschitz manifold is a Lipschitz ANR. This immediately follows from Theorem 5.1 in this paper.

Therefore what I said above works. It is enough to show that any two sufficiently $C^0$ close Lipschitz maps $f,g:M\to N$ are Lipschitz homotopic. Embed $N$ by a Lipschitz map into a Euclidean space $\mathbb R^n$, connect $f$ to $g$ by straight line homotopy in $\mathbb R^n$ and then use Lipschitz neighborhood retraction to push this homotopy into $N$.

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    $\begingroup$ The link goes through your university library. sciencedirect.com/science/article/abs/pii/S0166864120303205 is a better one. $\endgroup$ Feb 10, 2022 at 21:12
  • $\begingroup$ thanks Igor, I should have checked. It should be fixed now. $\endgroup$ Feb 10, 2022 at 21:54
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    $\begingroup$ Thanks. The last paper in particular seems to be a very good reference for all that relates to Lipschitz manifolds. $\endgroup$
    – Titti
    Feb 11, 2022 at 14:30

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