Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds
type: Item
instance of: scholarly article
title: Floer homology on the universal cover, Audin's conjecture and other constraints on Lagrangian submanifolds
zbMATH Open document ID: 1251.53049
DOI: 10.4171/CMH/259
author: Mihai Damian
published in: Commentarii Mathematici Helvetici
publication date: 2012-06-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1006.3398
Mathematics Subject Classification ID: 53D12
Mathematics Subject Classification ID: 53D40
Mathematics Subject Classification ID: 57R58
zbMATH DE Number: 6043065
zbMATH Keywords: Lagrangian embeddings
zbMATH Keywords: Floer homology
zbMATH Keywords: Maslov number
MaRDI profile type: Publication
Wikidata QID: Q123189916
arXiv ID: 1006.3398
review text: Let \((M^{2n}, \omega)\) be a symplectic manifold. A submanifold \(L^n\) of \(M\) is called \textit{Lagrangian} if the restriction of \(\omega\) to \(L\) vanishes. A fundamental question of symplectic geometry involves understanding necessary conditions for a manifold to admit a Lagrangian embedding in a given \(M\). The paper under review gives partial answers in the case of \textit{monotone} Lagrangian embeddings which are displaceable through Hamiltonian isotopies (this happens for instance when \(M = \mathbb{C}^n\)). The main result is that, if such \(L\) is aspherical, or more generally if the homology of its universal cover vanishes in odd degrees, its Maslov number \(N_L\) equals 2. The author also gives topological characterisations of Lagrangians \(L \hookrightarrow M\) with maximal Maslov number, in particular showing that if \(N_L = \dim(L) + 1\) then \(L\) is homeomorphic to a sphere, and if \(N_L = n \geq 6\) then \(L\) fibers over the circle and the fiber is homeomorphic to a sphere. A consequence is that any exact Lagrangian in \(T^*S^{2k+1}\) whose Maslov class is zero is homeomorphic to \(S^{2k+1}\).NEWLINENEWLINEThe main result thus extends the monotone version of `Audin's conjecture'; i.e. that an embedded Lagrangian torus in \( \mathbb{C}^n\) has Maslov number 2. While this conjecture remains wide open at the time of writing, the monotone case had received considerable previous attention, having been verified for \(n \leq 24\) in [\textit{Y.-G. Oh}, ``Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings'', Int. Math. Res. Not. 1996, No. 7, 305--346 (1996; Zbl 0858.58017)], and then for all \(n\) in [\textit{K. Fukaya}, \textit{Y.-G. Oh}, \textit{H. Ohta}, and \textit{K. Ono}, Lagrangian intersection Floer theory: anomaly and obstruction. Part I. AMS/IP Studies in Advanced Mathematics 46,1. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (2009; Zbl 1181.53002)]. The latter ref…
description: scientific article; zbMATH DE number 6043065