Smale, S.

What is the homotopy type of the group of diffeomorphisms of the 4-sphere? ★★★★

Author(s): Smale

Problem   $ Diff(S^4) $ has the homotopy-type of a product space $ Diff(S^4) \simeq \mathbb O_5 \times Diff(D^4) $ where $ Diff(D^4) $ is the group of diffeomorphisms of the 4-ball which restrict to the identity on the boundary. Determine some (any?) homotopy or homology groups of $ Diff(D^4) $.

Keywords: 4-sphere; diffeomorphisms

$C^r$ Stability Conjecture ★★★★

Author(s): Palis; Smale

Conjecture   Any $ C^r $ structurally stable diffeomorphism is hyperbolic.

Keywords: diffeomorphisms,; dynamical systems

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