Approximations of Lipschitz maps via immersions and differentiable exotic sphere theorems

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 dimM≤dimN. 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 Sn(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 Σn of general dimension n, there exists a bi-Lipschitz homeomorphism between Σn and Sn(1) which is a diffeomorphism except for a single point. In particular, there exists such a map between an exotic n-sphere Σn of dimension n>4 and Sn(1); the second one is that if an exotic 4-sphere Σ4 exists, then Σ4 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 Sn(1) which is a diffeomorphism except for one of points that attain their diameters.