关于扭结接触结构的三维辛化问题
The three-dimensional symplectization question for overtwisted contact structures
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中文总结 AI 辅助
该研究证明闭扭结接触三维流形的辛化辛同胚等价于其接触同胚，从而解决了接触结构均为扭结时的三维辛化问题。
中文摘要 AI 辅助
我们考虑带有正定向余定向接触结构的闭连通定向三维流形。我们证明，它们的辛化之间的辛同胚会迫使基础三维流形之间存在保定向微分同胚，使得两个接触结构作为定向平面场在此微分同胚下同伦。结合这种形式刚性与扭结接触结构的分类，我们证明两个闭扭结接触三维流形的辛化辛同胚当且仅当它们是接触同胚的。因此，当两个接触结构均为扭结时，我们解决了三维辛化问题。
英文摘要
We consider closed connected oriented three-manifolds equipped with positive cooriented contact structures. We prove that a symplectomorphism between their symplectizations forces the existence of an orientation-preserving diffeomorphism between the underlying three-manifolds under which the two contact structures are homotopic as oriented plane fields. Combining this formal rigidity with the classification of overtwisted contact structures, we show that two closed overtwisted contact three-manifolds have symplectomorphic symplectizations if and only if they are contactomorphic. Thus, we resolve the three-dimensional symplectization question when both contact structures are overtwisted.
