Кстати, опказывается гипотезу Смейла о том, что группа Diff(S^4) диффеоморфизмов
4-мерной сфера не гомотопически эквивалентна ортогональной группе O(4) вроде как доказали
и даже двумя способами

https://arxiv.org/abs/1812.02448

Some exotic nontrivial elements of the rational homotopy groups of Diff(S4)
Tadayuki Watanabe
This paper studies the rational homotopy groups of the group Diff(S4) of self-diffeomorphisms of S4 with the C∞-topology. We present a method to prove that there are many `exotic' non-trivial elements in π∗Diff(S4)⊗Q parametrized by trivalent graphs. As a corollary of the main result, the 4-dimensional Smale conjecture is disproved. The proof utilizes Kontsevich's characteristic classes for smooth disk bundles and a version of clasper surgery for families. In fact, these are analogues of Chern--Simons perturbation theory in 3-dimension and clasper theory due to Goussarov and Habiro.

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https://arxiv.org/abs/2008.07269

On the Smale Conjecture for Diff(S4)
Selman Akbulut
Recently Watanabe disproved the Smale Conjecture for S4, by showing Diff(S4)≠SO(5). He showed this by proving that their higher homotopy groups are different. Here we prove this more directly by showing π0Diff(S4)≠0, otherwise a certain loose-cork could not possibly be a loose-cork.