Volume 8, issue 2 (2004)

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Invariants for Lagrangian tori

Ronald Fintushel and Ronald J Stern

Geometry & Topology 8 (2004) 947–968
 arXiv: math.SG/0304402
Abstract

We define an simple invariant $\lambda \left(T\right)$ of an embedded nullhomologous Lagrangian torus and use this invariant to show that many symplectic 4–manifolds have infinitely many pairwise symplectically inequivalent nullhomologous Lagrangian tori. We further show that for a large class of examples that $\lambda \left(T\right)$ is actually a ${C}^{\infty }$ invariant. In addition, this invariant is used to show that many symplectic 4–manifolds have nontrivial homology classes which are represented by infinitely many pairwise inequivalent Lagrangian tori, a result first proved by S Vidussi for the homotopy K3–surface obtained from knot surgery using the trefoil knot.

Keywords
$4$–manifold, Seiberg–Witten invariant, symplectic, Lagrangian
Primary: 57R57
Secondary: 57R17