# Smale, S.

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

Author(s): Smale

\begin{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 $Diff(D^4)$. \end{problem}

Keywords: 4-sphere; diffeomorphisms

## $C^r$ Stability Conjecture ★★★★

Author(s): Palis; Smale

\begin{conjecture} Any $C^r$ structurally stable diffeomorphism is hyperbolic. \end{conjecture}

Keywords: diffeomorphisms,; dynamical systems