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Mohammad Ghomi
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Smale's theorem for $C^1$ diffeomorphisms of the sphere

In 1926 Kneser showed that homeomorphisms of $\mathbf{S}^2$ admit a retraction into the orthogonal group $O(3)$. Smale extended this result to Diffeomorphisms of $\mathbf{S}^2$ in 1958; however, in that paper he assumes that the diffeomorphisms are of class $C^k$ for $k\geq 2$. So my question is: Has anyone stablished Smale's theorem hold for $C^1$ diffeomorphisms?

Mohammad Ghomi
  • 7.1k
  • 1
  • 29
  • 54