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

Tadayuki Watanabe

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arXiv · 1812.02448

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

Tadayuki Watanabe

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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 $\pi_*\mathrm{Diff}(S^4)\otimes \mathbb{Q}$ 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.

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Tadayuki Watanabe. 2018-12-06. Some exotic nontrivial elements of the rational homotopy groups of $\mathrm{Diff}(S^4)$. https://arxiv.org/abs/1812.02448

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