4-dimensional Smale conjecture claims the following:


The inclusion $SO(5)$ → $SDiff(S^4)$ is a homotopy equivalence.


or Does $Diff(S^4)$ have the homotopy-type of $O(5)$ ?.


The inclusion $SO(n + 1$) → $SDiff(S^n)$ is a homotopy equivalence for n = 1 (trivial proof), n = 2 [1004,Smale,1959,Proc. Amer. Math. Soc.], n = 3 [464,Hatcher, 1983,Ann. of Math.], and is not a homotopy equivalence for n ≥ 5 [41,Antonelli, Burghelea, & Kahn,1972,Topology] and [164,Burghelea & Lashof,1974,Trans. Amer. Math. Soc.].


I looked everywhere but I could not find anything. Is this problem still open? Thanks.



What is the status of the 4-dimensional Smale Conjecture?
This problem is completely open.


