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Addendum to: Some exotic nontrivial elements of the rational homotopy groups of $mathrm{Diff}(S^4)$ (homological interpretation)

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 Added by Tadayuki Watanabe
 Publication date 2021
  fields
and research's language is English




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In this addendum, we give a differential form interpretation of the proof of the main theorem of arXiv:1812.02448, which gives lower bounds of the dimensions of $pi_k(Bmathrm{Diff}(D^4,partial))otimesmathbb{Q}$ in terms of the dimensions of Kontsevichs graph homology, and explain why it can be extended to arbitrary even dimensions $dgeq 4$. We attempted to make the proof accessible to more readers. Thus we do not assume familiarity with configuration space integrals nor knowledge of finite type invariants. Part of this addendum might be joined to the original article when it will be re-submitted to the journal. This is not aimed at giving a correction to the previous version.



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68 - Tadayuki Watanabe 2018
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 Kontsevichs 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.
170 - Andrew Putman 2019
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170 - Tadayuki Watanabe 2020
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