Some exotic nontrivial elements of the rational homotopy groups of $mathrm{Diff}(S^4)$

This paper studies the rational homotopy groups of the group $mathrm{Diff}(S^4)$ of self-diffeomorphisms of $S^4$ with the $C^infty$-topology. We present a method to prove that there are many `exotic non-trivial elements in $pi_*mathrm{Diff}(S^4)otimes mathbb{Q}$ parametrized by trivalent graphs. As a corollary of the main result, the 4-dimensional Smale conjecture is disproved. The proof utilizes Kontsevichs 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.