论文标题
拉格朗日填充物中拉格朗日手术的障碍
Obstructions to reversing Lagrangian surgery in Lagrangian fillings
论文作者
论文摘要
鉴于浸入式的Maslov- $ 0 $，如果填充物的索引和动作双点消失，则精确的Lagendrian结填充了Legendrian结的精确填充，然后通过Lagrangian手术，可以获得新的，Maslov-0 $，Maslov-$ 0 $，精确的Lagrangian填充物，较少的双重点，并增加了一个双重点。我们表明，并非总是有可能扭转Lagrangian手术：并非所有沉浸式的Maslov- $ 0 $，精确的Lagrangian填充物$ g \ geq 1 $和$ P $ Double点可以从这种Lagrangian手术中获得，以$ G-1 $填充$ g-1 $，并用$ P+1 $ $ $ double点获得。为了证明这一点，我们建立了沉浸式的Maslov- $ 0 $的存在之间的联系，具有Legendrian $λ$具有$ p $ p $ Double的Lagrangian填充，带有Action $ 0 $的$ P $ Double点，Maslov- $ 0 $的存在，Maslov- $ 0 $，Exact Lagrangian Cobortism cobortismiss cop $ p $ copies to $ P $ $ link $ hopf link to $ hopf link to $λ$λ$。然后，我们证明，增强量为嵌入式，Maslov- $ 0 $，Legendrian Links之间的Lagrangian Coobordisms的存在提供了阻碍。
Given an immersed, Maslov-$0$, exact Lagrangian filling of a Legendrian knot, if the filling has a vanishing index and action double point, then through Lagrangian surgery it is possible to obtain a new immersed, Maslov-$0$, exact Lagrangian filling with one less double point and with genus increased by one. We show that it is not always possible to reverse the Lagrangian surgery: not every immersed, Maslov-$0$, exact Lagrangian filling with genus $g \geq 1$ and $p$ double points can be obtained from such a Lagrangian surgery on a filling of genus $g-1$ with $p+1$ double points. To show this, we establish the connection between the existence of an immersed, Maslov-$0$, exact Lagrangian filling of a Legendrian $Λ$ that has $p$ double points with action $0$ and the existence of an embedded, Maslov-$0$, exact Lagrangian cobordism from $p$ copies of a Hopf link to $Λ$. We then prove that a count of augmentations provides an obstruction to the existence of embedded, Maslov-$0$, exact Lagrangian cobordisms between Legendrian links.
