Approximations of Lipschitz maps via immersions and differentiable exotic sphere theorems

TL;DR

This paper proves that Lipschitz maps between compact Riemannian manifolds can be smoothly approximated by immersions under certain conditions, leading to new differentiable sphere theorems and insights into exotic spheres and bi-Lipschitz homeomorphisms.

Contribution

It introduces a method to approximate Lipschitz maps by immersions when no singular points exist, and applies this to establish new differentiable sphere theorems including exotic spheres.

Findings

Bi-Lipschitz homeomorphisms without Clarke singular points are diffeomorphisms.

Existence of bi-Lipschitz maps between exotic spheres and standard spheres with single point diffeomorphism.

If an exotic 4-sphere exists, it must violate certain critical point conditions.

Abstract

As our main theorem, we prove that a Lipschitz map from a compact Riemannian manifold $M$ into a Riemannian manifold $N$ admits a smooth approximation via immersions if the map has no singular points on $M$ in the sense of F.H. Clarke, where $\dim M \leq \dim N$. As its corollary, we have that if a bi-Lipschitz homeomorphism between compact manifolds and its inverse map have no singular points in the same sense, then they are diffeomorphic. We have three applications of the main theorem: The first two of them are two differentiable sphere theorems for a pair of topological spheres including that of exotic ones. The third one is that a compact $n$-manifold $M$ is a twisted sphere and there exists a bi-Lipschitz homeomorphism between $M$ and the unit $n$-sphere $S^n(1)$ which is a diffeomorphism except for a single point, if $M$ satisfies certain two conditions with respect to critical points of its distance function in the Clarke sense. Moreover, we have three corollaries from the third theorem; the first one is that for any twisted sphere $\Sigma^n$ of general dimension $n$, there exists a bi-Lipschitz homeomorphism between $\Sigma^n$ and $S^n(1)$ which is a diffeomorphism except for a single point. In particular, there exists such a map between an exotic $n$-sphere $\Sigma^n$ of dimension $n>4$ and $S^n(1)$; the second one is that if an exotic $4$-sphere $\Sigma^4$ exists, then $\Sigma^4$ does not satisfy one of the two conditions above; the third one is that for any Grove-Shiohama type $n$-sphere $N$, there exists a bi-Lipschitz homeomorphism between $N$ and $S^n(1)$ which is a diffeomorphism except for one of points that attain their diameters.