Approximations of Lipschitz maps via immersions and differentiable exotic sphere theorems. (May 2017) - Kondo, Kei; Tanaka, Minoru
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 ≤ 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 Σ n of general dimension n, there exists a bi-Lipschitz homeomorphism between Σ 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 Σ n of dimension n > 4 and S n ( 1 ) ; the second one is that if an exotic 4 -sphere Σ 4 exists, then Σ 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.