Liouville hypersurfaces and connect sum cobordisms
type: Item
instance of: scholarly article
title: Liouville hypersurfaces and connect sum cobordisms
zbMATH Open document ID: 1491.53081
author: Russell Avdek
published in: The Journal of Symplectic Geometry
publication date: 2022-01-11T00:00:00Z
review text: In symplectic/contact geometry, the Moser/Gray trick to prove existence of the unique local model (up to symplectomorphism), constructions of isotopies, the flow of a natural vector field provide powerful tools to find nicer local shapes (or the model structure), to decompose a symplectic manifold into building blocks of the model structure, to glue pieces and to assemble topological/homological/categorical data. In [\textit{M. Abouzaid}, Ann. Math. (2) 175, No. 1, 71--185 (2012; Zbl 1244.53089)], the author defined a gluing operation for contact manifolds along nice neighborhoods of Liouville hypersurfaces, called the Liouville connected sum, which generalizes the Weinstein handle attachment for Weinstein (symplectic) manifolds. To do it, he introduced the concept of Liouville hypersurfaces (more generally, Liouville submanifolds) in contact manifolds which generalize ribbons of Legendrian graphs and pages of open book decomposition of contact manifolds. A fundamental construction is that any Liouville hypersurface \(\Sigma\) of a contact manifold can have a standard tubular neighborhood \(\mathcal{N}(\Sigma)\) with smooth, convex boundary, which can be considered as a flattening of \(\Sigma\) by the Reeb vector fields. A local model of Liouville connected sum is convex gluing of unit-cotangent bundles of two smooth compact manifolds with non-empty boundaries with orientation-reversing diffeomorphism on the boundary of the base manifolds (Example 2.14). A Liouville domain is a smooth compact symplectic manifold with boundary with exact symplectic form with Liouville vector field trasversely outward along the boundary. A Liouville hypersurface of a contact manifold \((M^{2n+1}, \xi\subset \text{Ker}(\alpha))\) is the image of an embedding \(i:\Sigma \to M\) of a Liouville domain \((\Sigma^{2n}, d\beta)\) of codimension \(1\) with \(i^{*}\alpha=\beta\). Since a Liouville hypersurface is transverse to the Reeb vector field \(R\) for \(\alpha\) (\(\alpha(R)=1\) and \(d…
reviewed by: Dahye Cho
Mathematics Subject Classification ID: 53D10
Mathematics Subject Classification ID: 57R90
Mathematics Subject Classification ID: 35B53
zbMATH DE Number: 7455583
zbMATH Keywords: Liouville connect sum
zbMATH Keywords: symplectic handle attachment
MaRDI profile type: Publication
DOI: 10.4310/JSG.2021.V19.N4.A2
description: scientific article; zbMATH DE number 7455583