Jongil Park
In this paper we construct a family of simply connected, spin, non-complex, symplectic 4-manifolds which cover all but finitely many allowed lattice points $(χ, c)$ lying in $0 \leq c \leq 8.76χ$. Furthermore, as a corollary, we prove that there exist infinitely many exotic smooth structures on $(2n+1)(S^2 \times S^2)$ for all n large enough.
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In this note, the geography problem in dimension four is reviewed and then its extension to dimension six for the symplectic case is explained. Finally some examples in dimension six are provided.
We produce simply connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides a symplectic extension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blow…
In this paper, results of J. Park and of B.D Park and Szabo on simply connected symplectic 4-manifolds are re-proven and extended to non-simply connected manifolds using Luttinger surgeries.
The geography of minimal symplectic 4-manifolds with arbitrary fundamental group and symplectic 6-manifolds with abelian fundamental group of small rank, and with arbitrary fundamental group are addressed.
In this article we consider a version of the geography question for simply-connected symplectic 4-manifolds that takes into account the divisibility of the canonical class as an additional parameter. We also find new examples of 4-manifolds admitting several symplectic structures, inequivalent under deformation and self-diffeomorphisms of the manifold.
The geography problem is usually stated for simply connected symplectic 4-manifolds. When the first cohomology is nontrivial, however, one can restate the problem taking into account how close the symplectic manifold is to satisfying the conclusion of the Hard Lefschetz Theorem, which is measured by a nonnegative integer called the degeneracy. In this paper we include the degeneracy as an extra parameter in the geography problem and show how to fill out the geography of symplectic 4-manifolds with Kodaira dimension 1 for all admissible triples.
We prove that for any contact 3-manifold supported by a spinal open book decomposition with planar pages, there is a universal bound on the Euler characteristic and signature of its minimal symplectic fillings. The proof is an application of the spine removal surgery operation recently introduced in joint work of the authors with Van Horn-Morris.
In this paper, we determine the Euler characteristics and signatures of the exact symplectic fillings of the contact double, 3-fold or 4-fold cyclic covers of the standard contact 3-sphere branched over certain transverse quasi-positive links. These links include all quasi-positive knots with crossing numbers smaller than 11 and all quasi-positive links with crossing numbers smaller than 12 and nonzero nullity.
Slope Inequalities for the Geography Problem of Spin Symplectic 4-Manifolds
该研究基于Roulleau与Urzúa的复曲面成果，构造了自旋辛4-流形的无限多互不同胚光滑结构，并指出其结果接近辛Bogomolov-Miyaoka-Yau不等式给出的最优边界。
In \cite{AP3, AHP}, the first author and his collaborators constructed the irreducible symplectic $4$-manifolds that are homeomorphic but not diffeomorphic to $(2n-1){\mathbb{CP}}^{2}\#(2n-1)\overline{\mathbb{CP}}^{2}$ for each integer $n \geq 25$, and the families of simply connected irreducible nonspin symplectic $4$-manifolds with positive signature that are interesting with respect to the symplectic geography problem. In this paper, we improve the main results in \cite{AP3, AHP}. In particular, we construct (i) an infinitely many irreducible symplectic and non-symplectic $4$-manifolds that…
The geography and botany of smooth/symplectic nonspin 4-manifolds with abelian fundamental group are addressed.
For any given finitely presented group G there exists a closed oriented 4-manifold with fundamental group G. What pairs of integers can occur as the signature and Euler characteristic of such a 4-manifold? The first part of this paper develops general theory related to this question and applies this theory to resolve the question for a large collection of groups, including surface groups and 3-manifold groups. The second part of the paper offers an extended analysis in the case of free abelian groups, G = Z^n. The paper concludes with a list of open problems related to the general question.
Aspherical 4-manifolds with positive Euler characteristic and their geography
本文显式构造了任意正整数n下满足χ=σ=n的闭定向非球面光滑4维流形，证明了Edmonds猜想，否定了非球面4维流形上的实Bogomolov-Miyaoka-Yau不等式，并回答了相关公开问题。
We construct symplectic surface bundles over surfaces with positive signatures for all but 18 possible pairs of fiber and base genera. Meanwhile, we determine the commutator lengths of a few new mapping classes.
Questions of geography of various classes of $4$-manifolds have been a central motivating question in $4$-manifold topology. Baykur and Korkmaz asked which small, simply connected, minimal $4$-manifolds admit a genus $2$ Lefschetz fibration. They were able to classify all the possible homeomorphism types and realize all but one with the exception of a genus $2$ Lefschetz fibration on a symplectic $4$-manifold homeomorphic, but not diffeomorphic to $3 \mathbb{CP}^2 \# 11\overline{\mathbb{CP}}^2$. We give a positive factorization of type $(10,10)$ that corresponds to such a genus $2$ Lefschetz f…
Geography of irreducible 4-manifolds with order two fundamental group
该研究针对具有二阶基本群的闭定向拓扑4-流形，通过对不可约单连通4-流形执行多种操作构建目标流形，证明除七个$(e, σ)$坐标外，满足特定条件的此类流形存在不可约光滑结构。
We discuss the geography problem of closed oriented 4-manifolds that admit a Riemannian metric of positive scalar curvature, and use it to survey mathematical work employed to address Gromov's observation that manifolds with positive scalar curvature tend to be inessential by focusing on the four-dimensional case. We also point out an strengthening of a result of Carr and its extension to the non-orientable realm.
Geography of pinched four-manifolds
本文研究了具有正或负夹挤截面曲率的有向四维流形，证明了其欧拉示性数和符号的新限制，并给出单连通情形下曲率介于0.161与1之间时流形同胚于S⁴或CP²的分类结果。
Symplectic forms on trisected 4-manifolds
本文研究闭光滑4-流形三分结构可构造辛结构的显式判据，结合此前辛流形存在相容三分的结论，给出4-流形容许辛结构的新刻画，并引出三维几何拓扑相关的若干研究问题。
In this paper we construct a family of symplectic 4--manifolds with positive signature for any given fundamental group $G$ that approaches the BMY line. The family is used to show that one cannot hope to do better than than the BMY inequality in finding a lower bound for the function $f=χ+bσ$ on the class of all minimal symplectic 4-manifolds with a given fundamental group.
Let X_1, X_2 be symplectic 4-manifolds containing symplectic surfaces F_1,F_2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X_1 and X_2 along the F_k. Using relative Gromov--Witten theory, we determine precisely when the symplectic 4-manifold Z is minimal (i.e., cannot be blown down); in particular, we prove that Z is minimal unless either: one of the X_k contains a (-1)-sphere disjoint from F_k; or one of the X_k admits a ruling with F_k as a section. As special cases, this proves a conjecture of Stipsicz asserting the minimality of fiber sums of Lefsche…
In this article we study proper symplectic and iso-symplectic embeddings of $4$--manifolds in $6$--manifolds. We show that a closed orientable smooth $4$--manifold admitting a Lefschetz fibration over $\C P^1$ admits a symplectic embedding in the symplectic manifold $(\C P^1 \times \C P^1 \times \C P^1, ω_{pr}),$ where $ω_{pr}$ is the product symplectic form on $\C P^1 \times \C P^1 \times \C P^1.$ We also show that there exists a sub-critical Weinstein $6$--manifold in which all finite type Weinstein $4$--manifolds admit iso-symplectic embeddings.
We introduce a surgery operation on symplectic manifolds called coisotropic Luttinger surgery, which generalizes Luttinger surgery on Lagrangian tori in symplectic 4-manifolds. We use it to produce infinitely many distinct symplectic non-Kahler 6-manifolds $X$ with $c_1(X)=0$ which are not of the form $M\times F$ for $M$ a symplectic 4-manifold and $F$ a closed surface.
We answer a question of Oprea-Tralle on the realizability of symplectic algebras by symplectic manifolds in dimensions divisible by four, along with a question of Lupton-Oprea in all even dimensions. This will also allow us to address, in all even dimensions six and higher, another question of Oprea-Tralle on the possibility of algebraic conditions on the rational homotopy minimal model of a closed smooth manifold implying the existence of a symplectic structure on the manifold.
We mostly determine which closed smooth oriented 4-manifolds fibering over lower dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4-manifolds.
A symplectic form is called hyperbolic if its pull-back to the universal cover is a differential of a bounded one-form. The present paper is concerned with the properties and constructions of manifolds admitting hyperbolic symplectic forms. The main results are: * If a symplectic form represents a bounded cohomology class then it is hyperbolic. * The symplectic hyperbolicity is equivalent to a certain isoperimetric inequality. * The fundamental group of symplectically hyperbolic manifold is non-amenable. We also construct hyperbolic symplectic forms on certain bundles and Lefschetz fibrations,…
We exhibit many examples of closed symplectic manifolds on which there is an autonomous Hamiltonian whose associated flow has no nonconstant periodic orbits (the only previous explicit example in the literature was the torus T^2n (n\geq 2) with an irrational symplectic structure). The underlying smooth manifolds of our examples include, for instance: the K3 surface and also infinitely many smooth manifolds homeomorphic but not diffeomorphic to it; infinitely many minimal four-manifolds having any given finitely-presented group as their fundamental group; and simply connected minimal four-manif…
We show that recent results of Friedl-Vidussi and Chen imply that a symplectic manifold admits a fixed point free circle action if and only if it admits a symplectic circle action and we give a complete description of the symplectic cone in this case. This then completes the characterisation of symplectic 4-manifolds that admit non-trivial circle actions.
This article analyzes the interplay between symplectic geometry in dimension four and the invariants for smooth four-manifolds constructed using holomorphic triangles introduced in math.SG/0110169. Specifically, we establish a non-vanishing result for the invariants of symplectic four-manifolds, which leads to new proofs of the indecomposability theorem for symplectic four-manifolds and the symplectic Thom conjecture. As a new application, we generalize the indecomposability theorem to splittings of four-manifolds along a certain class of three-manifolds obtained by plumbings of spheres. This …
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.
We consider the existence of symplectic and conformal symplectic codimension-one foliations on closed manifolds of dimension at least 5. Our main theorem, based on a recent result by Bertelson-Meigniez, states that in dimension at least 7 any almost contact structure is homotopic to a conformal symplectic foliation. In dimension 5 we construct explicit conformal symplectic foliations on every closed, simply-connected, almost contact manifold, as well as honest symplectic foliations on a large subset of them. Lastly, via round-connected sums, we obtain, on closed manifolds, examples of conforma…
We show that a generic Hamiltonian diffeomorphism on a closed symplectic manifold which is symplectically aspherical has at least the stable Morse number of fixed points - this is in line with a conjecture by Arnold.
We prove that a positive definite smooth four-manifold with $b_2^+ \geq 2$ and having either no 1-handles or no 3-handles cannot admit a symplectic structure.
In this paper we construct a minimal symplectic 4-manifold and prove it is homeomorphic but not diffeomorphic to CP^2 # 3(-CP^2)
Withdrawn due to an incompleteness of the main results.
We verify that the rational blow-down schemes along certain Seifert fibered 3-manifolds found by the second author, Szabo and Wahl are, in fact, symplectic operations.
We show that a stabilized convex symplectic (also called Liouville) manifold with the homotopy type of a half dimensional CW-complex is symplectomorphic to a flexible Weinstein manifold.
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…
The purpose of this paper is to present some results on the existence of homologous, nonisotopic symplectic or lagrangian surfaces embedded in a simply connected symplectic 4-dimensional manifold.
We study the question of how many embedded symplectic or Lagrangian tori can represent the same homology class in a simply connected symplectic 4-manifold.
We describe Lefschetz-Bott fibrations on complex line bundles over symplectic manifolds explicitly. As an application, we construct more than one strong symplectic filling of the link of the $A_{k}$-type singularity. In the appendix, we show that the total space of a Lefschetz-Bott fibration over the unit disk serves as a strong symplectic filling of a contact manifold compatible with an open book induced by the fibration.
Almost toric manifolds form a class of singular Lagrangian fibered symplectic manifolds that is a natural generalization of toric manifolds. Notable examples include the K3 surface, the phase space of the spherical pendulum and rational balls useful for symplectic surgeries. The main result of the paper is a complete classification up to diffeomorphism of closed almost toric four-manifolds. A key step in the proof is a geometric classification of the singular affine structures that can occur on the base of a closed almost toric four-manifold.
A method of computation of its terms is presented together with some stabilization results. As an application a characterization of symplectic harmonic manifolds is given and a relationship with the C-spectral sequence is indicated.
We construct a new family {K_n} of simply connected symplectic 4-manifolds with the property c_1^2(K_n)/chi(K_n) -> 9 (as n goes to infinity).
In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP^2#3(-CP^2) which contains a symplectic surface of genus 2, trivial normal bundle, and simply connected complement and a disjoint nullhomologous Lagrangian torus with the fundamental group of the complement generated by one of the loops on the torus. (2) A minimal symplectic 4-manifold that is homeomorphic but …
This is the first of a series of papers devoted to the topology of symplectic Calabi-Yau $4$-manifolds endowed with certain symplectic finite group actions. We completely determine the fixed-point set structure of a finite cyclic action on a symplectic Calabi-Yau $4$-manifold with $b_1>0$. As an outcome of this fixed-point set analysis, the $4$-manifold is shown to be a $T^2$-bundle over $T^2$ in some circumstances, e.g., in the case where the group action is an involution which fixes a $2$-dimensional surface in the $4$-manifold. Our project on symplectic Calabi-Yau $4$-manifolds is based on …
We study the number of Darboux charts needed to cover a closed connected symplectic manifold $(M,ω)$, and effectively estimate this number from below and from above in terms of the Lusternik--Schnirelmann category of $M$ and the Gromov width of $(M,ω)$.
In \cite{btoric}, Guillemin et al. proved a Delzant-type theorem which classifies $b$-symplectic toric manifolds. More generally, in \cite{torus} they proved a similar convexity result for general Hamiltonian torus action on $b$-symplectic manifolds. In this paper, we provide a new way to construct $b$-symplectic toric manifolds from usual toric manifolds. Conversely, through this way, we can also decompose a $b$-symplectic toric manifolds to usual toric manifolds. Finally, we will try to prove that this kind of decomposition is useful, although the symplectic structure for our decomposition o…
Let M be a closed oriented smooth 4-manifold admitting symplectic structures. If M is minimal and has b^+=1, we prove that there is a unique symplectic canonical class up to sign, and any real second cohomology class of positive square is represented by symplectic forms. Similar results hold when M is not minimal.
In this article we construct a minimal symplectic 4-manifold R that has small Euler characteristic (e(R)=8) and two essential Lagrangian tori with nice properties. These properties make R particularly suitable for constructing interesting examples of symplectic manifolds with small Euler characteristic. In particular, we construct an exotic symplectic CP^2# 5(-CP^2), the smallest known minimal symplectic 4-manifold with pi_1=Z, the smallest known minimal symplectic 4-manifolds with pi_1=Z/a + Z/b for all a,b in Z, and the smallest known minimal symplectic 4-manifold with pi_1=Z^3. We use the p…
In this article we apply the technique of Luttinger surgery to study the complexity of the fundamental group of symplectic $4$-manifolds with holomorphic Euler number $χ_h=1$. We discuss the topology of symplectic $4$-manifolds with $b^+=1$ and provide various constructions of symplectic $4$-manifolds with $b^+=1$ and prescribed $c_1^2$.
In this paper, we construct simply connected symplectic Calabi-Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along $T^4$. Using our construction, we also produce symplectic non-Kähler Calabi-Yau 6-manifolds with fundamental group $\Z$. In this paper, we also produce the first examples of simply connected symplectic Calabi-Yau and non-Calabi-Yau 6-manifolds via coisotropic Luttinger surgery.
In this paper we classify symplectic Lefschetz fibrations (with empty base locus) on a four-manifold which is the product of a three-manifold with a circle. This result provides further evidence in support of the following conjecture regarding symplectic structures on such a four-manifold: if the product of a three-manifold with a circle admits a symplectic structure, then the three-manifold must fiber over a circle, and up to a self-diffeomorphism of the four-manifold, the symplectic structure is deformation equivalent to the canonical symplectic structure determined by the fibration of the t…
Symplectic 4-manifolds with Kodaira dimension zero can be viewed as symplectic Calabi-Yau surfaces. We are able to completely determine their Betti numbers by proving two general results on quaternionic vector bundles.
This is a survey paper on the space of symplectic structures on closed 4-manifolds, for the Proceedings ICCM 2004
On totally c-symplectically aspherical manifolds
该研究在≥6的偶数维构造了具非平凡第二同伦群的全辛非球面凯勒流形例子，在四维构造了具非平凡第二同伦群的全c-辛非球面近辛流形例子，拓展了相关流形的存在性结果。
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