How many smooth structures a manifold can have? This question turns out to be a deep one, and built a new branch in geometry. For some results people got the Fields medal:

  • There exists topological manifolds with no smooth structure. The first example was a compact 10-dimensional manifold found by Kervaire [1]. According to Lee in [2], for every \(n\ge 4\) there exists a compact \(n\)-manifold that have no smooth structure at all.
  • If a topological manifold admits a smooth structure, it has uncountably many distinct ones [2, Problem 1.6]. However, the picture changes if we identify smooth structures by diffeomorphisms .
  • A topological manifold of dimension \( n \le 3 \) has exactly one smooth structure up to diffeomorphism (Munkers, 1960; Moise, 1977).
  • The flat space \(\mathbb{R}^n\), for \(n\neq 4\), has only the standard smooth structure for every \(n\neq 4\). On \(\mathbb{R}^4\) there exists uncountably many smooth structures which are not diffeomorphic to each other (Donaldson-Freedman, 1984). The Euclidean space with nonstandard structures are called fake \(\mathbb{R}^4\)’s.
  • For \(n\le 6\) and \(n\neq 4\), \(\mathbb{S}^n\) has only the standard smooth structure up to diffeomorphism. For \(n=7\) there are 15 (Milnor, 1956).

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References Link to heading

  1. M. Kervaire, A manifold which does not admit any differentiable structure, Commentarii Mathematici Helvetici, vol. 34, p. 257–270, 1960. doi:10.1007/BF02565940
  2. J. Lee, Introduction to Smooth Manifolds. New York ; London: Springer, 2013.