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

Tadayuki Watanabe

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

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arxiv 1812.02448 v3 pith:W7342WN5 submitted 2018-12-06 math.GT math.AT

Tadayuki Watanabe This is my paper · ORCID

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keywords diffmathrmclasperelementsexoticgroupshomotopyrational

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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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Exotic families of embeddings
math.GT 2025-01 conditional novelty 6.0 of 10

Smooth embeddings of 3-manifolds in 4-manifolds that are topologically trivial but smoothly exotic, both as individual embeddings and in families parameterized by spheres, are constructed and detected.

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