Non-orientable Lagrangian surfaces in rational 4-manifolds

Geometry & Topology, 2012

Given a Lagrangian sphere in a symplectic 4-manifold .M; !/ with b C D 1, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension Ä of .M; !/ is 1, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is not characteristic, we show that homologous Lagrangian spheres are smoothly isotopic, and when the Euler number is less than 8, we generalize Hind and Evans' Hamiltonian uniqueness in the monotone case. On the existence side, when Ä D 1, we give a characterization of classes represented by Lagrangian spheres, which enables us to describe the non-Torelli part of the symplectic mapping class group.

Asian Journal of Mathematics, 2002

In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular, we completely determine which classes with square at least -1 in such manifolds can be represented by embedded spheres. Moreover, based on a new characterization of the action of the diffeomorphism group on the intersection forms of a rational manifold, we are able to determine the orbits of the diffeomorphism group on the set of classes represented by embedded spheres of square at least -1 in any 4-manifold admitting a symplectic structure.

arXiv (Cornell University), 2019

Let (X, ω) be a symplectic rational surface. We study the space of tamed almost complex structures Jω using a fine decomposition via smooth rational curves and a relative version of the infinite dimensional Alexander-Pontrjagin duality. This decomposition provides new understandings of both the variation and stability of the symplectomorphism group Symp(X, ω) when deforming ω. In particular, we compute the rank of π 1 (Symp(X, ω)) with χ(X) ≤ 7 in terms of the number Nω of (-2)-symplectic sphere classes.

2012

We show that, for any two orientable smooth open 4-manifolds X 0 , X 1 which are homeomorphic, their cotangent bundles T˚X 0 , T˚X 1 are symplectomorphic with their canonical symplectic structure. In particular, for any smooth manifold R homeomorphic to R 4 , the standard Stein structure on T˚R is Stein homotopic to the standard Stein structure on T˚R 4 " R 8. We use this to show that any exotic R 4 embeds in the standard symplectic R 8 as a Lagrangian submanifold. As a corollary, we show that R 8 has uncountably many smoothly distinct foliations by Lagrangian R 4 s with their standard smooth structure.

Proceedings of the American Mathematical Society

We characterize rational or ruled surfaces among all symplectic 4-manifolds by the existence of certain smoothly embedded spheres.

Geometry and Topology of Manifolds, 2005

In this paper we show that every degree 2 homology class of a 2n-dimensional symplectic manifold is represented by an immersed symplectic surface if it has positive symplectic area. Moreover, the symplectic surface can be chosen to be embedded if 2n is at least 6. We also analyze the additional conditions under which embedded symplectic representatives exist in dimension 4.

Geometric and Functional Analysis, 2021

Let ω denote an area form on S 2 . Consider the closed symplectic 4manifold M = (S 2 × S 2 , Aω ⊕ aω) with 0 < a < A. We show that there are families of displaceable Lagrangian tori L 0,x , L 1,x ⊂ M , for x ∈ [0, 1], such that the twocomponent link L 0,x ∪ L 1,x is non-displaceable for each x.

arXiv: Geometric Topology, 2020

We show that any closed oriented 3-manifold can be topologically embedded in some simply-connected closed symplectic 4-manifold, and that it can be made a smooth embedding after one stabilization. As a corollary of the proof we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exists 3-manifolds which cannot smoothly embed in a way that appropriately respect orientations in any symplectic manifold with weakly convex boundary. This embedding obstruction can also be used to detect exotic smooth structures on 4-manifolds.

2000

We present a new proof of a result due to Taubes: if X is a closed symplectic four-manifold with b_+(X) &gt; 1+b_1(X) and with some positive multiple of the symplectic form a rational class, then the Poincare dual of the canonical class of X may be represented by an embedded symplectic submanifold. The result builds on the existence of Lefschetz

Mathematische Zeitschrift, 2010

We study symplectic surfaces in ruled symplectic 4-manifolds which are disjoint from a given symplectic section. As a consequence, in any symplectic 4-manifold, two homologous symplectic surfaces which are C 0 close must be Hamiltonian isotopic.