Linking class of Lagrangian or totally real embeddings
type: Item
instance of: scholarly article
title: Linking class of Lagrangian or totally real embeddings
zbMATH Open document ID: 0939.37034
DOI: 10.1023/A:1006519728625
author: Vincent Borrelli
published in: Annals of Global Analysis and Geometry
publication date: 2000-07-03T00:00:00Z
review text: Given any totally real embedding \(j\) of the torus \({\mathbb T}^2\) into \({\mathbb C}^2\) and any nonvanishing vector field \(v\) in \({\mathbb T}^2\), one can construct a 1-class \(\sigma_1(j,v) \in H^1({\mathbb T}^2,{\mathbb Z})\) called the linking class of \(j\) in the direction \(v\). The class is an effective tool to detect arcwise connected components of the space of totally real embeddings \(\text{EmbTR}({\mathbb T}^2,{\mathbb C}^2)\) which are not detected by the Maslov class. In the first part of the present paper, the construction of this linking class is generalized to any closed, oriented manifold \(M^n\) admitting a totally real embedding \(j\) into \({\mathbb C}^n\) . With every totally real embedding \(j:M^n \to {\mathbb C}^n\) and with every nonvanishing vector field \(v\) on \(M^n\) the author associates an element \(\sigma_{n-1}(j,v) \in H^{n-1}(M^n,{\mathbb Z})\), its linking class. As in the case of the torus \({\mathbb T}^2\), the linking class \(\sigma_{n-1}\) allows the detection of arcwise connected components of the space of totally real embeddings \(\text{EmbTR}(M^n,{\mathbb C}^n)\). In the second part of the paper, the author shows the nontriviality of his linking class by building families of totally real embeddings into \({\mathbb C}^n\) of product manifolds of the form \(M^{n-1} \times S^1\).
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 1340154
zbMATH Keywords: totally real embeddings
zbMATH Keywords: linking class
zbMATH Keywords: Lagrangian
reviewed by: Vagn Lundsgaard Hansen
MaRDI profile type: Publication
full work available at URL: a:1006519728625
OpenAlex ID: W160075387
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description: scientific article; zbMATH DE number 1340154