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arXiv · 2207.13205

Obstructions to reversing Lagrangian surgery in Lagrangian fillings

Orsola Capovilla-Searle·Noémie Legout·Maÿlis Limouzineau·Emmy Murphy·Yu Pan·Lisa Traynor
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Abstract

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 $\Lambda$ 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 $\Lambda$. We then prove that a count of augmentations provides an obstruction to the existence of embedded, Maslov-$0$, exact Lagrangian cobordisms between Legendrian links.

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Orsola Capovilla-Searle, Noémie Legout, Maÿlis Limouzineau, Emmy Murphy, Yu Pan, Lisa Traynor. 2022-07-26. Obstructions to reversing Lagrangian surgery in Lagrangian fillings. https://arxiv.org/abs/2207.13205

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