%0 Generic %1 watanabe2018exotic %A Watanabe, Tadayuki %D 2018 %K conjecture smale %T Some exotic nontrivial elements of the rational homotopy groups of $Diff(S^4)$ %U http://arxiv.org/abs/1812.02448 %X This paper studies the rational homotopy groups of the group $Diff(S^4)$ of self-diffeomorphisms of $S^4$ with the $C^ınfty$-topology. We present a method to prove that there are many `exotic' non-trivial elements in $\pi_*Diff(S^4)Q$ 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.

@misc{watanabe2018exotic, 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.}, added-at = {2019-10-12T17:59:04.000+0200}, author = {Watanabe, Tadayuki}, biburl = {https://www.bibsonomy.org/bibtex/2a0b777bddc71595d6fb6658942a7882e/gzhou}, description = {Some exotic nontrivial elements of the rational homotopy groups of $\mathrm{Diff}(S^4)$}, interhash = {3c7011d30f0f414c6826f5083e12cf32}, intrahash = {a0b777bddc71595d6fb6658942a7882e}, keywords = {conjecture smale}, note = {cite arxiv:1812.02448Comment: 74 pages, 27 figures. Added Appendix B}, timestamp = {2019-10-12T17:59:04.000+0200}, title = {Some exotic nontrivial elements of the rational homotopy groups of $\mathrm{Diff}(S^4)$}, url = {http://arxiv.org/abs/1812.02448}, year = 2018 }