On Stein fillings of the 3-torus T^3

Infinitely many small exotic Stein fillings

We show that there exist infinitely many simply connected compact Stein 4-manifolds with b_2=2 such that they are all homeomorhic but mutually non-diffeomorphic, and they are Stein fillings of the same contact 3-manifold on their boundaries. We also describe their handlebody pictures.

具有无穷多个斯坦因填充的平面接触三维流形

Planar contact 3-manifolds with infinitely many Stein fillings

研究由平面开书支撑的闭接触三维流形，通过证明其有无穷多个两两非同胚的斯坦因填充，回答了K3问题4.105，并得出存在特定接触三维流形的相关推论。

Stein fillings of homology $3$-spheres and mapping class groups

In this article, using combinatorial techniques of mapping class groups, we show that a Stein fillable integral homology $3$-sphere supported by an open book decomposition with page a $4$-holed sphere admits a unique Stein filling up to diffeomorphism. Furthermore, according to a property of deforming symplectic fillings of a rational homology $3$-spheres into strongly symplectic fillings, we also show that a symplectically fillable integral homology $3$-sphere supported by an open book decomposition with page a $4$-holed sphere admits a unique symplectic filling up to diffeomorphism and blow-…

Strongly fillable contact 3-manifolds without Stein fillings

We use the Ozsvath-Szabo contact invariant to produce examples of strongly symplectically fillable contact 3-manifolds which are not Stein fillable.

Factorizations in SL(2,Z) and simple examples of inequivalent Stein fillings

We give simple examples of elements of SL(2,Z) admitting inequivalent factorizations into products of Dehn twists. This can be interpreted in terms of inequivalent Stein fillings of a same contact 3-manifold by genus 1 Lefschetz fibrations over the disk.

Stein fillings of contact 3-manifolds obtained as Legendrian surgeries

In this note, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on $(S^3,ξ_{std})$ along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an infinite family of contact 3-manifolds which can be obtained by Legendrian surgeries on $(S^3,ξ_{std})$ along certain Legendrian twist knots. As a corollary, we obtain a classification of Stein fillings of an infinite family of contact hyperbolic 3-manifolds up to symplectic deforma…

Indefinite Stein fillings and Pin(2)-monopole Floer homology

Given a spin$^c$ rational homology sphere $(Y,\mathfrak{s})$ with $\mathfrak{s}$ self-conjugate and for which the reduced monopole Floer homology $\mathit{HM}_{\bullet}(Y,\mathfrak{s})$ has rank one, we provide obstructions to the intersection forms of its Stein fillings which are not negative definite. The proof of this result (and of its natural generalizations we discuss) uses $\mathrm{Pin}(2)$-monopole Floer homology.

Singularity links with exotic Stein fillings

In a recent paper of Akhmedov, Etnyre, Mark and Smith, it was shown that there exist infinitely many contact Seifert fibered 3-manifolds each of which admits infinitely many exotic (homeomorphic but pairwise non-diffeomorphic) simply-connected Stein fillings. Here we extend this result to a larger set of contact Seifert fibered 3-manifolds with many singular fibers and observe that these 3-manifolds are singularity links. In addition, we prove that the contact structures induced by the Stein fillings are the canonical contact structures on these singularity links. As a consequence, we verify a…

Families of contact 3-manifolds with arbitrarily large Stein fillings

We show that there are vast families of contact 3-manifolds each member of which admits infinitely many Stein fillings with arbitrarily big euler characteristics and arbitrarily small signatures ---which disproves a conjecture of Stipsicz and Ozbagci. To produce our examples, we set a framework which generalizes the construction of Stein structures on allowable Lefschetz fibrations over the 2-disk to those over any orientable base surface, along with the construction of contact structures via open books on 3-manifolds to spinal open books introduced in [24].

Mapping class group relations, Stein fillings, and planar open book decompositions

The aim of this paper is to use mapping class group relations to approach the `geography' problem for Stein fillings of a contact 3-manifold. In particular, we adapt a formula of Endo and Nagami so as to calculate the signature of such fillings as a sum of the signatures of basic relations in the monodromy of a related open book decomposition. We combine this with a theorem of Wendl to show that for any Stein filling of a contact structure supported by a planar open book decomposition, the sum of the signature and Euler characteristic depends only on the contact manifold. This gives a simple o…

Stein fillable Seifert fibered 3-manifolds

We characterize the closed, oriented, Seifert fibered 3-manifolds which are oriented boundaries of Stein manifolds. We also show that for this class of 3-manifolds the existence of Stein fillings is equivalent to the existence of symplectic fillings.

Topological complexity of symplectic 4-manifolds and Stein fillings

We prove that there exists no a priori bound on the Euler characteristic of a closed symplectic 4-manifold coming solely from the genus of a compatible Lefschetz pencil on it, nor is there a similar bound for Stein fillings of a contact 3-manifold coming from the genus of a compatible open book --- except possibly for a few low genera cases. To obtain our results, we produce the first examples of factorizations of a boundary parallel Dehn twist as arbitrarily long products of positive Dehn twists along non-separating curves on a fixed surface with boundary. This solves an open problem posed by…

Stein fillings and SU(2) representations

We recently defined invariants of contact 3-manifolds using a version of instanton Floer homology for sutured manifolds. In this paper, we prove that if several contact structures on a 3-manifold are induced by Stein structures on a single 4-manifold with distinct Chern classes modulo torsion then their contact invariants in sutured instanton homology are linearly independent. As a corollary, we show that if a 3-manifold bounds a Stein domain that is not an integer homology ball then its fundamental group admits a nontrivial homomorphism to SU(2). We give several new applications of these resu…

Contact 3-manifolds with infinitely many Stein fillings

Infinitely many contact 3-manifolds each admitting infinitely many, pairwise non-diffeomorphic Stein fillings are constructed. We use Lefschetz fibrations in our constructions and compute their first homologies to distinguish the fillings.

Stein 填充与 Milnor 纤维的比较

Stein fillings vs. Milnor fibers

本文比较正规曲面奇点链环的 Stein 填充与 Milnor 填充，证明 Milnor 填充具有有界拓扑且对 sandwiched 奇点仅有有限多个，并构造了非来自 Milnor 纤维或消解的 Stein 填充。

无编织线路图用于棱镜空间的Stein填充

Unbraided wiring diagrams for Stein fillings of lens spaces

本文提出了一种无编织线路图算法，用于描述棱镜空间的Stein填充与环形商奇点的Milnor纤维之间的双射关系。

On Stein fillings of contact torus bundles

We consider a large family F of torus bundles over the circle, and we use recent work of Li--Mak to construct, on each Y in F, a Stein fillable contact structure C. We prove that (i) each Stein filling of (Y,C) has vanishing first Chern class and first Betti number, (ii) if Y in F is elliptic then all Stein fillings of (Y,C) are pairwise diffeomorphic and (iii) if Y in F is parabolic or hyperbolic then all Stein fillings of (Y,C) share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles Y in F we exhibit non-homotopy…

On symplectic fillings of virtually overtwisted torus bundles

We use Menke's JSJ-type decomposition theorem for symplectic fillings to reduce the classification of strong and exact symplectic fillings of virtually overtwisted torus bundles to the same problem for tight lens spaces. For virtually overtwisted structures on elliptic or parabolic torus bundles, this gives a complete classification. For virtually overtwisted structures on hyperbolic torus bundles, we show that every strong or exact filling arises from a filling of a tight lens space via round symplectic 1-handle attachment, and we give a condition under which distinct tight lens space filling…

Strong symplectic fillability of contact torus bundles

In this paper, we study strong symplectic fillability and Stein fillability of some tight contact structures on negative parabolic and negative hyperbolic torus bundles over the circle. For the universally tight contact structure with twisting $π$ in $S^1$-direction on a negative parabolic torus bundle, we completely determine its strong symplectic fillability and Stein fillability. For the universally tight contact structure with twisting $π$ in $S^1$-direction on a negative hyperbolic torus bundle, we give a necessary condition for it being strongly symplectically fillable. For the virtually…

Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres

A conjecture due to Gompf asserts that no nontrivial Brieskorn homology sphere admits a pseudoconvex embedding in ${\mathbb C}^2$, with either orientation. A related question asks whether every compact contractible 4-manifold admits the structure of a Stein domain. We verify Gompf's conjecture, with one orientation, for a family of Brieskorn spheres of which some are known to admit a smooth embedding in ${\mathbb C}^2$. With the other orientation our methods do not resolve the question, but do give rise to an example of a contractible, boundary-irreducible 4-manifold that admits no Stein struc…

Exact Lagrangian Fillings of Legendrian $(2,n)$ torus links

For a Legendrian $(2,n)$ torus knot or link with maximal Thurston-Bennequin number, Ekholm, Honda, and Kálmán constructed $C_n$ exact Lagrangian fillings, where $C_n$ is the $n$-th Catalan number. We show that these exact Lagrangian fillings are pairwise non-isotopic through exact Lagrangian isotopy. To do that, we compute the augmentations induced by the exact Lagrangian fillings $L$ to $\mathbb{Z}_2[H_1(L)]$ and distinguish the resulting augmentations.

非预期Stein填充、有理曲面奇点与平面曲线配置

Unexpected Stein fillings, rational surface singularities, and plane curve arrangements

针对约化基本闭链的有理曲面奇点，研究构建辛几何类比构造，通过辛圆盘配置描述其链环的所有Stein填充，证明部分奇点存在非Milnor纤维的Stein填充，同时给出极小消解自交数≤-5时Stein填充唯一的结论。

Exotic Stein fillings with arbitrary fundamental group

For any finitely presentable group $G$, we show the existence of an isolated complex surface singularity link which admits infinitely many exotic Stein fillings such that the fundamental group of each filling is isomorphic to $G$. We also provide an infinite family of closed exotic smooth four-manifolds with the fundamental group $G$ such that each member of the family admits a non-holomorphic Lefschetz fibration over the two-sphere.

关于三叶结的手术与辛填充

Surgeries on the trefoil and symplectic fillings

本文确定了对右手三叶结进行特定手术后所得流形上接触结构的Stein可填充性，发现其可填充与否缺乏明显规律。

Fillings of unit cotangent bundles

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $Σ_g$, where $g$ is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $Σ_g$. For exact fillings, we show that the rational homology agrees with that of the disk cotangent bundle, and that the integral homology takes on finitely many possible values: for example, if $g-1$ is square-free, then any exact filling has the same integral homology and intersection fo…

Homology spheres bounding acyclic smooth manifolds and symplectic fillings

In this paper, we collect various structural results to determine when an integral homology $3$--sphere bounds an acyclic smooth $4$--manifold, and when this can be upgraded to a Stein manifold. In a different direction we study whether smooth embedding of connected sums of lens spaces in $\mathbb{C}^2$ can be upgraded to a Stein embedding, and determined that this never happens.

The diffeomorphism type of symplectic fillings

We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a degree-theoretic analysis of the evaluation map on a suitable moduli space of holomorphic spheres. Applications of this homological result include a proof that compositions of right-handed Dehn twists on Liouville domains are of infinite order in the symplectomorphism group. We a…

Small exotic Stein manifolds

It is known that the only Stein filling of the standard contact structure on S^3 is B^4. In this paper, we construct simply connected exotic compact Stein 4-manifold pairs for any Betti number $b_2 \geq 1$; we do this by enlarging corks and plugs.

关于脊柱开书分解的辛填充 II：全纯曲线与分类

On symplectic fillings of spinal open book decompositions II: Holomorphic curves and classification

本文证明当接触3流形具有平面页的统一脊柱开书分解时，其(弱、强和/或精确)辛和Stein填充可由Lefschetz纤维化的微分同胚类分类。进一步结果包括新的ECH接触不变量和SFT中代数扭力消失准则，非定向圈 bundles的填充分类，以及关于四维实数中Stein结构变形类的

Genus Zero One Point Correlators of Subcritical Stein Manifolds

We determine the one point genus zero correlators of compactly supported forms of a subcritical Stein filling whose first Chern class vanishes. This is a step towards determining the full potential function of the filling. As an application, we proved that if a Kähler manifold $M^{2n}$ admits a subcritical polarization and $c_1$ vanishes in the subcritical complement, then $M$ is uniruled.

Correlators and Descendants of Subcritical Stein Manifolds

We determine contact homology algebra of a subcritical Stein-fillable contact manifold whose first Chern class vanishes. We also compute the genus-0 one point correlators and gravitational descendants of compactly supported closed forms of their subcritical Stein fillings. This is a step towards determining the full potential function of the filling as defined in \cite{EliashbergGiventalHofer}. These invariants also give a canonical presentation of the cylindrical contact homology. With respect to this presentation, we determine the degree-2 differential in the Bourgeois--Oancea exact sequence…

关于脊柱开放书分解的辛填充 I：几何构造

On symplectic fillings of spinal open book decompositions I: Geometric constructions

本文介绍了脊柱开放书与接触结构及辛或Stein结构的关系，提出新的辛cobordism构造脊柱移除手术，扩展了Eliashberg、Gay-Stipsicz和第三作者的构造，并应用该手术得到新的非强（有时非弱）辛填充的接触流形例子。

Classification of multiplicity free Hamiltonian actions of complex tori on Stein manifolds

A Hamiltonian action of a complex torus on a symplectic complex manifold is said to be {\it multiplicity free} if a general orbit is a lagrangian submanifold. To any multiplicity free Hamiltonian action of a complex torus $T\cong (\C^\times)^n$ on a Stein manifold $X$ we assign a certain 5-tuple consisting of a Stein manifold $Y$, an étale map $Y\to \t^*$, a set of divisors on $Y$ and elements of $H^2(Y,\Z)^{\oplus n}, H^2(Y,\C)$. We show that $X$ is uniquely determined by this invariants. Furthermore, we describe all 5-tuples arising in this way.

Circular spherical divisors and their contact topology

This paper investigates the symplectic and contact topology associated to circular spherical divisors. We classify, up to toric equivalence, all concave circular spherical divisors $ D $ that can be embedded symplectically into a closed symplectic 4-manifold and show they are all realized as symplectic log Calabi-Yau pairs if their complements are minimal. We then determine the Stein fillability and rational homology type of all minimal symplectic fillings for the boundary torus bundles of such $D$. When $ D $ is anticanonical and convex, we give explicit betti number bounds for Stein fillings…

Non-fillable augmentations of twist knots

We establish new examples of augmentations of Legendrian twist knots that cannot be induced by orientable Lagrangian fillings. To do so, we use a version of the Seidel-Ekholm-Dimitroglou Rizell isomorphism with local coefficients to show that any Lagrangian filling point in the augmentation variety of a Legendrian knot must lie in the injective image of an algebraic torus with dimension equal to the first Betti number of the filling. This is a Floer-theoretic version of a result from microlocal sheaf theory. For the augmentations in question, we show that no such algebraic torus can exist.

扭转纺丝环面链环的精确拉格朗日填充

Exact Lagrangian fillings of twist-spun torus links

本文构造了在旋转对称下保持不变的勒让德环面链环的精确拉格朗日填充，并利用组合方法将其推广到扭转纺丝环面，核心贡献是给出了对称最大弱分离集存在的充要条件。

Analytic filling of totally real tori

We prove that any embedded Maslov index two analytic disc attached to a totally real torus in the complex two-dimensional affine space extends to an analytic filling provided that the torus is contained in a regular level set of a strictly plurisubharmonic function.

可容许相对Stein对上的相容相对Lefschetz纤维化

Compatible Relative Lefschetz Fibrations On Admissible Relative Stein Pairs

本文将四维球面Lefschetz纤维化相关结果推广至六维情形，构造了以(p,q,2)-Brieskorn簇为页的六维球面相容Lefschetz纤维化，并在具可容许相对Stein对结构的六维Stein域上构建相对Lefschetz纤维化，给出纯拓扑存在性证明。

Generalized Chain Surgeries and Applications

We describe the Stein handlebody diagrams of Milnor fibers of Brieskorn singularities $x^p + y^q + z^r = 0$. We also study the natural symplectic operation by exchanging two Stein fillings of the canonical contact structure on the links in the case $p = q = r$, where one of the fillings comes from the minimal resolution and the other is the Milnor fiber. We give two different interpretations of this operation, one as a symplectic sum and the other as a monodromy substitution in a Lefschetz fibration.

接触边界无法决定的Stein结构

Stein structures not determined by the contact boundary

该研究构造了接触边界与第一陈类一致但辛形式不同构的Stein结构，解决了K3问题列表中相对填充问题的归一化版本。

Berge duals and universally tight contact structures

Dehn surgery on a knot determines a dual knot in the surgered manifold, the core of the filling torus. We consider duals of knots in $S^3$ that have a lens space surgery. Each dual supports a contact structure. We show that if a universally tight contact structure is supported, then the dual is in the same homology class as the dual of a torus knot.

具有Calabi-Yau填充的平面接触结构与拓扑量子计算

Planar Contact Structures with Calabi-Yau Fillings and Topological Quantum Computation

本文研究平面开书的Calabi-Yau填充判据及其在Ising表示中的量子操作，证明CY因子分解长度最小化拓扑量，并揭示接触拓扑与量子纠缠的关联。

Envelopes of holomorphy and holomorphic discs

The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a nei…

Strongly Fillable Contact Manifolds and J-holomorphic Foliations

We prove that every strong symplectic filling of a planar contact manifold admits a symplectic Lefschetz fibration over the disk, and every strong filling of the 3-torus similarly admits a Lefschetz fibration over the annulus. It follows that strongly fillable planar contact structures are also Stein fillable, and all strong fillings of the 3-torus are equivalent up to symplectic deformation and blowup. These constructions result from a compactness theorem for punctured J-holomorphic curves that foliate a convex symplectic manifold. We use it also to show that the compactly supported symplecto…