Nonisotopic Symplectic Tori in the Fiber Class of Elliptic Surfaces

Homologous Non-isotopic Symplectic Tori in Homotopy Rational Elliptic Surfaces

Let E(1)_K denote the closed 4-manifold that is homotopy equivalent (hence homeomorphic) to the rational elliptic surface E(1) and is obtained by performing Fintushel-Stern knot surgery on E(1) using a knot K in S^3. We construct an infinite family of homologous non-isotopic symplectic tori representing a primitive homology class in E(1)_K when K is any nontrivial fibred knot in S^3. We also show how these tori can be non-isotopically embedded as homologous symplectic submanifolds in other symplectic 4-manifolds.

Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves

We discuss the properties of a certain type of Dehn surgery along a Lagrangian torus in a symplectic 4-manifold, known as Luttinger's surgery, and use this construction to provide a purely topological interpretation of a non-isotopy result for symplectic plane curves with cusp and node singularities due to Moishezon.

Monodromy groups of Lagrangian tori in the symplectic 4-space

We determine the Lagrangian monodromy group L(T) and the smooth monodromy group S(T) of a Clifford torus T in the symplectic 4-space. We show that L(T) is isomorphic to the infinite dihedral group, and S(T) is generated by three reflections. We give explicit formulas for both groups. We also show that if a Lagrangian torus is smoothly isotopic to a Clifford torus then the smooth isotopy can be chosen to be Lagrangian outside of a disc.

切触手术图中的混合环面

Mixed tori in contact surgery diagrams

本研究提出将辛JSJ分解应用于三维切触流形切触手术图的图示框架，可推导原流形的恰当/弱辛填充，既重现已知分类结果，又给出大类配边三维流形的填充分类算法。

Monodromy and isotopy of monotone Lagrangian tori

We define new Hamiltonian isotopy invariants for a monotone Lagrangian torus embedded in a symplectic 4-manifold. We show that, in the standard symplectic 4-space, these invariants distinguish a monotone Clifford torus from a Chekanov torus.

Elliptic diffeomorphisms of symplectic 4-manifolds

We show that symplectically embedded $(-1)$-tori give rise to certain elements in the symplectic mapping class group of $4$-manifolds. An example is given where such elements are proved to be of infinite order.

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$. In a different direction, we prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered then the 4-manifold is virtually symplectic unless its virtual first Betti number is 1.

Invariants for Lagrangian tori

We define an simple invariant 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(T) is actually a C-infinity 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 …

环面上的圆模式空间及其辛形式

Space of circle patterns on tori and its symplectic form

本文研究闭环面上圆模式空间的辛几何与拓扑结构，证明其通过嵌入获得的 Weil-Petersson 辛形式具有非退化性，并证明该空间同胚于闭环面的 Teichmüller 空间。