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)?
Best Answer
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 $N$.