Linking class of Lagrangian or totally real embeddings
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 \(M^{n-1} \times S^1\).
scientific article; zbMATH DE number 1340154