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Maslov class of exact Lagrangians and cylindrical handles

Axel Husin

TL;DR

This work shows that the Maslov class need not vanish for closed exact Lagrangians after adding cylindrical or critical handles to Weinstein domains, contrasting the Abouzaid–Kragh result for cotangent bundles of closed manifolds. The authors introduce cylindrical handles as a generalization of Weinstein handles and provide a local model in which attaching a cylindrical handle yields a closed exact Lagrangian with nonzero Maslov class for some extension of the quadratic complex volume form on $W$. They prove Theorems 1.1 and 1.2 via a two-step strategy: (i) explicit constructions in the model $D^{2n}$ and (ii) a localization argument that transfers the obstruction to arbitrary Weinstein domains via contact Darboux balls. These results show the subcritical-handle hypothesis in previous work is essential and connect Lagrangian mutations and surgeries to Maslov-class obstructions.

Abstract

A fundamental and deep result in symplectic topology due to Abouzaid and Kragh states that the Maslov class vanishes for closed exact Lagrangians in cotangent bundles of closed manifolds. In this article we prove by an explicit construction that the Maslov class does not vanish in general for closed exact Lagrangians in Weinstein domains obtained by performing a critical handle attachment to a cotangent bundle. We also define cylindrical handles as a generalization of Weinstein handles.

Maslov class of exact Lagrangians and cylindrical handles

TL;DR

This work shows that the Maslov class need not vanish for closed exact Lagrangians after adding cylindrical or critical handles to Weinstein domains, contrasting the Abouzaid–Kragh result for cotangent bundles of closed manifolds. The authors introduce cylindrical handles as a generalization of Weinstein handles and provide a local model in which attaching a cylindrical handle yields a closed exact Lagrangian with nonzero Maslov class for some extension of the quadratic complex volume form on . They prove Theorems 1.1 and 1.2 via a two-step strategy: (i) explicit constructions in the model and (ii) a localization argument that transfers the obstruction to arbitrary Weinstein domains via contact Darboux balls. These results show the subcritical-handle hypothesis in previous work is essential and connect Lagrangian mutations and surgeries to Maslov-class obstructions.

Abstract

A fundamental and deep result in symplectic topology due to Abouzaid and Kragh states that the Maslov class vanishes for closed exact Lagrangians in cotangent bundles of closed manifolds. In this article we prove by an explicit construction that the Maslov class does not vanish in general for closed exact Lagrangians in Weinstein domains obtained by performing a critical handle attachment to a cotangent bundle. We also define cylindrical handles as a generalization of Weinstein handles.

Paper Structure

This paper contains 5 sections, 2 theorems, 31 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Cylindrical handles
  3. Construction in D2n U H
  4. Proof of Theorem 1.1
  5. Proof of Theorem 1.2

Key Result

Theorem 1.1

Let $W$ be a Weinstein domain with $\dim W=2n\geq 6$ and with a compatible almost complex structure and a quadratic complex volume form. Then one can to $W$ first attach a Weinstein one-handle $H_1$ and then attach a critical Weinstein handle $H_n$ so that the quadratic complex volume form on $W$ ex

Figures (1)

  • Figure 1: The Lagrangian $L\cup K\subset W\cup H$

Theorems & Definitions (3)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3