Some exotic nontrivial elements of the rational homotopy groups of $\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}(S^4)$

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