# Trivalent graphs and diffeomorphisms of disks

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I will explain a geometric method to construct families of diffeomorphisms of manifolds by using a higher dimensional analogue of Goussarov-Habiro's trivalent graph surgery in 3-dimension. This would produce lots of potentially nontrivial families of diffeomorphisms of manifolds. In particular, our construction gives that the homotopy groups $\pi_k \mathrm{Diff}_{\partial}(D^n) \otimes \mathbb{Q}$ are non-trivial for many $k$ and for all dimensions $n\geq 4$. These non-trivialities can be detected by Kontsevich's configuration space integrals. In dimension 4, we obtain a negative answer to the 4-dimensional analogue of the Smale conjecture. Namely, the group $\mathrm{Diff}_{\partial}(D^4)$ is not contractible.