Annals of Global Analysis and Geometry

, Volume 17, Issue 4, pp 371–384 | Cite as

Linking Class of Lagrangian or Totally Real Embeddings

  • Vincent Borrelli


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 σn− 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 ℂ n 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: Hn− 1 (M n , ℤ) → Hn− 1 (M n , ℤ2) are the same (if n even).

embedding Lagrangian embedding linking totally real embedding 


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Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Vincent Borrelli
    • 1
  1. 1.Institut Girard Desargues -- UPRES-A 5028 du CNRSUniversité Claude BernardVilleurbanne CedexFrance

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