Linking Class of Lagrangian or Totally Real Embeddings

Abstract

Given a totally real embedding j of the 2-torus $\mathbb{T}^{\text{2}}$ into ℂ², one defines a 1-class σ₁ – its linking class – which is a tool to detect arcwise connected components of the space of totally real embeddings EmbTr($\mathbb{T}^{\text{2}}$, ℂ²). We generalize the construction of the linking class to any totally real embedding j of a connected, oriented, compact manifold without boundary M ⁿ into ℂⁿ. We obtain an (n − 1)-class σ _{n− 1} which is still an invariant for isotopy classes of totally real embeddings. We show that this class is nontrivial by computing it for some families of totally real embeddings. We then study the relationship between isotopy classes of ordinary embeddings and the linking class. With additional assumptions on M ⁿ (n ≥ 4 and M ⁿ parallelizable) we obtain the following: two totally real embeddings of M ⁿ into ℂⁿ which belong to the same isotopy class of totally real immersion, belong to the same isotopy class of ordinary embedding if and only if (1) their linking classes are the same (if n odd); (2) the images of their linking classes by the coefficient homomorphism μ: H ^{n− 1} (M ⁿ, ℤ) → H ^{n− 1} (M ⁿ, ℤ₂) are the same (if n even).