The Group of Contactomorphisms of the Sphere Fixing an Overtwisted Disk

Examples of non-trivial contact mapping classes for overtwisted contact manifolds in all dimensions

We construct (infinitely many) examples in all dimensions of contactomorphisms of closed overtwisted contact manifolds that are smoothly isotopic but not contact-isotopic to the identity.

Non-loose unknots, overtwisted discs, and the contact mapping class group of $S^3$

We classify Legendrian unknots in overtwisted contact structures on $S^3$. In particular, we show that up to contact isotopy for every pair $(n,\pm(n-1))$ with $n>0$ there are exactly two oriented non-loose Legendrian unknots in $S^3$ with Thurston-Bennequin invariant $n$ and rotation number $\pm(n-1)$. (Only one overtwisted contact structure on $S^3$ admits a non-loose unknot $K$ and the classical invariants have to be $\mathrm{tb}(K)=n$ and $\mathrm{rot}(K)=\pm(n-1)$ for $n>1$.) This can be used to prove two results attributed to Y.~Che\-kan\-ov: The first implies that the contact mapping cl…

相对H-原理与切触几何

Relative H-Principle and Contact Geometry

本文证明满足相对h-原理的完整解空间到形式解空间的映射存在同伦截面，并将其应用于过扭曲切触结构，在切触同胚群的同伦群中构造出无限循环子群。