A Lagrangian Klein bottle you can't squeeze
type: Item
instance of: scholarly article
title: A Lagrangian Klein bottle you can't squeeze
DOI: 10.1007/978-3-031-19111-4_17
author: Jonathan D. Evans
publication date: 2024-09-11T00:00:00Z
review text: If \(L\) is a closed, orientable, Lagrangian surface in a symplectic 4-manifold \((X,\omega)\), then the genus of \(L\) is \(g(L)=1+\tfrac{1}{2}[L]^2\) and any family of symplectic forms \(\omega_t\), with \(\omega_0=\omega\), on \(X\) with respect to which \(L\) is smoothly isotopic to a Lagrangian must satisfy \(\int_L[\omega_t]=0\) for all \(t\). If, on the other hand, \(L\) is a nonorientable Lagrangian surface, then \(H^2(L;\mathbb{R})=0\) and the topology of \(L\) admits a less rigid relationship with \([\omega]\). This observation leads to the following two guiding questions for the present paper:\N\NQuestion 1.1. Given a symplectic 4-manifold \((X,\omega)\) and a \(\mathbb{Z}/2\)-homology class \(\beta\in H_2(X;\mathbb{Z}/2)\), what is the minimal nonorientable genus of a nonorientable Lagrangian surface \(L\subset X\) with \([L]=\beta\)?\N\NQuestion 1.2. Given a symplectic 4-manifold \((X,\omega)\) and a nonorientable Lagrangian surface \(L\subset X\), how far can you deform \(\omega\) in cohomology before there is no Lagrangian smoothly isotopic to \(L\)?\N\NThe 4-manifold for which the paper is addressing these questions is \(X=S^2\times S^2\), where all symplectic forms are diffeomorphic to one of the form\N\[\N\omega = \lambda\,p_1^*\sigma + p_2^*\sigma,\N\]\Nwhere \(p_1,p_2\colon X\to S^2\) are the obvious projections, \(\sigma\in\Omega^2(S^2)\) is an area form, and \(\lambda>0\) is a constant. Letting \(\beta\) be the \(\mathbb{Z}/2\)-homology class \([\{*\}\times S^2]\), a Lagrangian Klein bottle \(L\) representing \(\beta\) is constructed whenever \(\lambda<2\). Namely, \(L\) is constructed as a visible Lagrangian and also as a tropical Lagrangian in the almost toric 4-manifold \((X,\omega)\). Because this construction breaks down when \(\lambda\geq 2\), the author conjectures that \(\beta\) cannot be represented by a Lagrangian Klen bottle when \(\lambda\geq 2\), a conjecture whose proof would provide an answer to the minimal genus question above.\…
reviewed by: Austin Christian
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 53D35
Mathematics Subject Classification ID: 53D42
zbMATH DE Number: 7910419
zbMATH Keywords: Lagrangian Klein bottles
zbMATH Keywords: symplectic 4-manifolds
zbMATH Keywords: pseudoholomorphic curves
MaRDI profile type: Publication
cites work: Normal bundles of double dimension immersions, double points of Lagrangian immersions, and totally real embeddings
cites work: Compactness results in symplectic field theory
cites work: Minimal Lagrangian surfaces in \(\mathbb S^2 \times \mathbb S^2\)
cites work: Nonorientable Lagrangian surfaces in rational 4-manifolds
cites work: Lagrangian imbeddings of surfaces and unfolded Whitney umbrella
cites work: Pseudo holomorphic curves in symplectic manifolds
cites work: Examples of tropical-to-Lagrangian correspondence
cites work: Homology class of a Lagrangian Klein bottle
cites work: Lagrangian Klein bottles in \({\mathbb{R}^{2n}}\)
cites work: The surgery of Lagrange submanifolds
cites work: Lagrangian embeddings of the Klein bottle and combinatorial properties of mapping class groups
cites work: Symplectic triangle inequality
cites work: Q4459923
description: scientific article; zbMATH DE number 7910419