Some general constructions of tight mappings

Synopsis

We present several general methods of constructing tight hypersurfaces f: Mⁿ → Eⁿ⁺¹, n ≧ 2. We prove a smoothing lemma, which allows us to approximate tight continuous hypersurfaces by C^∞ tight ones. We show that given a tight immersion or C^∞-stable map f: M² → E³ of a compact surface M² other than S², there is a tight C∞-stable map g: M² # RP²:→ E³. We prove that given C^∞ tight immersions f: Mⁿ → Eⁿ⁺¹ and g: Nⁿ → Eⁿ⁺¹ of compact n-manifolds Mⁿ and Nⁿ into Eⁿ⁺¹, there is a tight C^∞ immersion of Mⁿ # Nⁿ into Eⁿ⁺¹. Two other methods involve hypersurfaces of rotation and sets in Eⁿ⁺¹ at a fixed distance from a tightly embedded n-manifold with boundary in Eⁿ. One consequence of these methods is that the outer part of C^∞ tight hypersurfacesf: Mⁿ → Eⁿ⁺¹, n ≧ 3, is far more complicated than in the case of tight surfaces in Eⁿ. For example, given any C^∞n-manifold M with boundary tightly embedded in Eⁿ, there is a tight immersion f:Nⁿ → Eⁿ⁺¹ of a closed n-manifold having M as a topset. Kuiper's theorem describing tight immersions of surfaces into E³ does not generalise to the case of hypersurfaces f: Mⁿ → Eⁿ⁺¹, n ≧ 3, without substantial restrictions on Mⁿ and/or f.