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

Abstract

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 ${\pi }_{\ast }\mathrm{Diff}(S^4)\otimes \mathbb{Q}$ parametrized by trivalent graphs. As a corollary of the main result, the 4-dimensional Smale conjecture is disproved. The proof utilizes 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.

74 pages, 27 figures. Added Appendix B

References in corpus (2)

• Tropical curves, graph complexes, and top weight cohomology of M_g
• On the Kontsevich-Kuperberg-Thurston construction of a configuration-space invariant for rational homology 3-spheres

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• On an infinite order cork automorphisms
• The Geometry of Loop Spaces III: Isometry Groups of Contact Manifolds
• Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $\mathrm{Diff}(S^4)$ (homological interpretation)