Topologically distinct Lagrangian and symplectic fillings
type: Item
instance of: scholarly article
title: Topologically distinct Lagrangian and symplectic fillings
zbMATH Open document ID: 1305.57044
author: Chang Cao
author: Nathaniel Gallup
author: Kyle Hayden
author: Joshua M. Sabloff
published in: Mathematical Research Letters
publication date: 2014-09-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1307.7998
review text: In some cases, the smooth topology of a symplectic manifold is defined by the contact topology of its boundary. For example, in [Lond. Math. Soc. Lect. Note Ser. 151, 45--72 (1990; Zbl 0731.53036)], \textit{Y.~Eliashberg} proved that any symplectically aspherical filling of the standard contact \(S^3\) is diffeomorphic to \(B^4\). \textit{B. Chantraine} in [Algebr. Geom. Topol. 10, No. 1, 63--85 (2010; Zbl 1203.57010)], showed that if \(L\) is an orientable Lagrangian filling of an oriented Legendrian link \(\Lambda\) in the boundary of a Stein surface, then \(\text{tb}(\Lambda)=-\chi(L)\), and if \(\Lambda\) is a knot, then the topology of a Lagrangian filling of \(\Lambda\) is completely determined by its Thurston-Bennequin invariant \(\text{tb}(\Lambda)\). Based on this result, in this paper the authors are interested if the topology of a Lagrangian filling of an oriented Legendrian link in \(S^3\) is completely determined by its Thurston-Bennequin invariant. The authors show that, in contrast to the the case of Legendrian knots, the topology is not defined by the tb-invariant, and they show that for every integer \(N>1\), there exists a Legendrian link \(\Lambda\subset(S^3,\xi_0)\) with \(p(N)\) non-homeomorphic exact orientable Lagrangian fillings, where \(p(N)\) denotes the number of partitions of \(N\). Also, they show that for every integer \(N>1\), there exists a link \(K\subset S^3\) with \(p(N)\) non-singular complex algebraic curves that intersect \(S^3\) transversally in \(K\) with the property that the intersections of the curves with \(B^4\) are pairwise non-homeomorphic. Finally, the authors extend the first result to higher dimensions and they show that for all integers \(n\), \(N>1\), there exists a connected Legendrian submanifold \(\Lambda\subset(S^{2n+1},\xi_0)\) with \(p(N)\) non-homeomorphic connected exact orientable Lagrangian fillings.
reviewed by: Andrew Bucki
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 53D12
zbMATH DE Number: 6350066
zbMATH Keywords: symplectic filling
zbMATH Keywords: Lagrangian filling
zbMATH Keywords: Legendrian link
zbMATH Keywords: contact 3-manifold
MaRDI profile type: Publication
arXiv ID: 1307.7998
DOI: 10.4310/MRL.2014.V21.N1.A7
Recommended article: Subcritical contact surgeries and the topology of symplectic fillings
Recommended article: Symplectic Fillings, Contact Surgeries, and Lagrangian Disks
Recommended article: Surfaces in the 4-disk with the same boundary and fundamental group
Recommended article: Non-displaceable contact embeddings and infinitely many leaf-wise intersections
Recommended article: Lagrangian fillings of Legendrian 4-plat knots
Recommended article: On the topology of fillings of contact 3-manifolds
Recommended article: Lagrangian fillings and complicated Legendrian unknots
Recommended article: Fillings of unit cotangent bundles of nonorientable surfaces
Recommended article: Legendrian contact homology in 𝑃×ℝ
Recommended article: Lifting pseudo-holomorphic polygons to the symplectisation of \(P \times \mathbb{R}\) and applications
description: scientific article