We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashberg's h-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.
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.
We make some elementary observations concerning subcritically Stein fillable contact structures on 5-manifolds. Specifically, we determine the diffeomorphism type of such contact manifolds in the case the fundamental group is finite cyclic, and we show that on the 5-sphere the standard contact structure is the unique subcritically fillable one. More generally, it is shown that subcritically fillable contact structures on simply connected 5-manifolds are determined by their underlying almost contact structure. Along the way, we discuss the homotopy classification of almost contact structures.
We continue our study of contact structures on manifolds of dimension at least five using complex surgery theory. We show that in each dimension 2q+1 > 3 there are 'maximal' almost contact manifolds to which there is a Stein cobordism from any other (2q+1)-dimensional contact manifold. We show that the product M x S^2 admits a weakly fillable contact structure provided M admits a weak symplectic filling. We also study the connection between Stein fillability and connected sums: we give examples of almost contact manifolds for which the connected sum is Stein fillable, while the components are …
A two-dimensional open book (S,h) determines a closed, oriented three-manifold Y(S,h) and a contact structure C(S,h) on Y(S,h). The contact structure C(S,h) is Stein fillable if h is positive, i.e. h can be written as a product of right-handed Dehn twists. Work of Wendl implies that when S has genus zero the converse statement holds, that is if C(S,h) is Stein fillable then h is positive. On the other hand, results by Wand and by Baker, Etnyre and Van Horn-Morris imply the existence of counterexamples to the converse statement with S of arbitrary genus strictly greater than one. The main purpo…
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-…
For any integer $n\geq 2$, we construct an infinite family of Stein fillable contact $(4n-1)$-manifolds each of which admits infinitely many pairwise homotopy inequivalent Stein fillings.
We construct a family of Stein fillable contact homology 3-spheres such that each contact structure of the family is supported by an open book with planar page, and a Stein filling of the contact manifold is of Mazur type.
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…
For a 4-manifold represented by a framed knot in $S^3$, it has been well known that the 4-manifold admits a Stein structure if the framing is less than the maximal Thurston-Bennequin number of the knot. In this paper, we prove either the converse of this fact is false or there exists a compact contractible oriented smooth 4-manifold (with Stein fillable boundary) admitting no Stein structure. Note that an exotic smooth structure on $S^4$ exists if and only if there exists a compact contractible oriented smooth 4-manifold with $S^3$ boundary admitting no Stein structure.
In this article, we prove a generalization of a theorem of Lisca-Matic to Stein cobordisms and develop a method for distinguishing certain Stein cobordisms using rotation numbers. Using these results along with standard techniques from convex surface theory and classifications of tight contact structures on certain 3-manifolds due to Honda, we classify the tight contact structures on a certain class of plumbed 3-manifolds that bound non-simply connected 4-manifolds. Moreover, we give descriptions of the Stein fillings of the Stein fillable contact structures.
We introduce a new generalization of Gompf nuclei and give applications. We construct infinitely many exotic smooth structures for a large class of compact 4-manifolds with boundary, regarding topological invariants. We prove that a large class of closed 3-manifolds (including disjoint unions of Stein fillable 3-manifolds) bound compact connected oriented 4-manifolds which admit infinitely many smooth structures. To detect smooth structures, we introduce a relative genus function. As a side result, we show that log transform and knot surgery of 4-manifolds do not admit any Stein structure, und…
An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Nemethi. We also show that the construction cannot always work in the reverse direction: in fact the U-filtration depth of contact Ozsvath-Szabo invariant obstructs the exist…
According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an application, we derive classification results for subcritically Stein fillable contact 5-manifolds and characterise the standard contact structure on the 5-sphere in terms of such fillings. This characterisation is discussed in the context of the Andrews-Curtis conjecture concern…
In this note we observe that one can contact embed all contact 3-manifolds into a Stein fillable contact structure on the twisted $S^3$-bundle over $S^2$ and also into a unique overtwisted contact structure on $S^3\times S^2$. These results are proven using "spun embeddings" and Lefschetz fibrations.
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…
This is a survey on contact open books and contact Dehn surgery. The relation between these two concepts is discussed, and various applications are sketched, e.g. the monodromy of Stein fillable contact 3-manifolds, the Giroux-Goodman proof of Harer's conjecture on fibred links, construction of symplectic caps to fillings (Eliashberg, Etnyre), and detection of non-loose Legendrian knots with the help of contact surgery.
We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set $\mathbb{Z}_{\geq0}\cup\{\infty\}$. It is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsváth-Szabó contact class.
We define an invariant of contact 3-manifolds with convex boundary using Kronheimer and Mrowka's sutured instanton Floer homology theory. To the best of our knowledge, this is the first invariant of contact manifolds -- with or without boundary -- defined in the instanton Floer setting. We prove that our invariant vanishes for overtwisted contact structures and is nonzero for contact manifolds with boundary which embed into Stein fillable contact manifolds. Moreover, we propose a strategy by which our contact invariant might be used to relate the fundamental group of a closed contact 3-manifol…
Two of the basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. We present the first such classification on an infinite family of (mostly) hyperbolic 3-manifolds: surgeries on the figure-eight knot. We also determine which of the tight contact structures are symplectically fillable and which are universally tight.
Our main result gives an adjunction inequality for embedded surfaces in certain $4$-manifolds with contact boundary under a non-vanishing assumption on the Bauer--Furuta type invariants. Using this, we give infinitely many knots in $S^3$ that are not smoothly H-slice (that is, bounding a null-homologous disk) in many $4$-manifolds but they are topologically H-slice. In particular, we give such knots in the boundaries of the punctured elliptic surfaces $E(2n)$. In addition, we give obstructions to codimension-0 orientation-reversing embedding of weak symplectic fillings with $b_3=0$ into closed…
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…
Building on work of Stolz, we prove for integers $0 \le d \le 3$ and $k>232$ that the boundaries of $(k-1)$-connected, almost closed $(2k+d)$-manifolds also bound parallelizable manifolds. Away from finitely many dimensions, this settles longstanding questions of C.T.C. Wall, determines all Stein fillable homotopy spheres, and proves a conjecture of Galatius and Randal-Williams. Implications are drawn for both the classification of highly connected manifolds and, via work of Kreck and Krannich, the calculation of their mapping class groups. Our technique is to recast the Galatius and Randal-Wi…
Given a 4-manifold X and an imbedding of T^{2} x B^2 into X, we describe an algorithm X --> X_{p,q} for drawing the handlebody of the 4-manifold obtained from X by (p,q)-logarithmic transforms along the parallel tori. By using this algorithm, we obtain a simple handle picture of the Dolgachev surface E(1)_{p,q}, from that we deduce that the exotic copy E(1)_{p,q} # 5(-CP^2) of E(1) # 5(-CP^2) differs from the original one by a codimension zero simply connected Stein submanifold M_{p,q}, which are therefore examples of infinitely many Stein manifolds that are exotic copies of each other (rel bo…
Loi and Piergallini showed that a smooth compact, connected $4$-manifold $X$ with boundary admits a Stein structure if and only if $X$ is a simple branched cover of a $4$-disk $D^4$ branched along a positive braided surface $S$ in a bidisk $D_{1}^{2} \times D_{2}^{2} \approx D^4$. For each integer $N \geq 2$, we construct a braided surface $S_{N}$ in $D^4$ and simple branched covers $X_{N, 1}, X_{N, 2}, \dots , X_{N, N}$ of $D^{4}$ branched along $S_{N}$ such that the covers have the same degrees, and they are mutually diffeomorphic, but the Stein structures associated to the covers are mutual…
We introduce symplectic Calabi-Yau caps to obtain new obstructions to exact fillings. In particular, it implies that any exact filling of the standard unit cotangent bundle of a hyperbolic surface has vanishing first Chern class and has the same integral homology and intersection form as its disk cotangent bundle. This gives evidence to a conjecture that all of its exact fillings are diffeomorphic to the disk cotangent bundle. As a result, we also obtain the first infinitely family of Stein fillable contact 3-manifolds with uniform bounds on the Betti numbers of its exact fillings but admittin…
In this paper, we show that the Ozsváth-Szabó contact invariant $c^+(ξ)\in HF^+(-Y)$ of a contact 3-manifold $(Y,ξ)$ can be calculated combinatorially if $Y$ is the boundary of a certain type of plumbing $X$, and $ξ$ is induced by a Stein structure on $X$. Our technique uses an algorithm of Ozsváth and Szabó to determine the Heegaard-Floer homology of such 3-manifolds. We discuss two important applications of this technique in contact topology. First, we show that it simplifies the calculation of the Ozsváth-Stipsicz-Szabó obstruction to admitting a planar open book. Then we define a numerical…
We initiate a study of positive multisections of Lefschetz fibrations via positive factorizations in framed mapping class groups of surfaces. Using our methods, one can effectively capture various interesting symplectic surfaces in symplectic 4-manifolds as multisections, such as Seiberg-Witten basic classes and exceptional classes, or branched loci of compact Stein surfaces as branched coverings of the 4-ball. Various problems regarding the topology of symplectic 4-manifolds, such as the smooth classification of symplectic Calabi-Yau 4-manifolds, can be translated to combinatorial problems in…
In this article, we study the maximal length of positive Dehn twist factorizations of surface mapping classes. In connection to fundamental questions regarding the uniform topology of symplectic 4-manifolds and Stein fillings of contact 3-manifolds coming from the topology of supporting Lefschetz pencils and open books, we completely determine which boundary multitwists admit arbitrarily long positive Dehn twist factorizations along nonseparating curves, and which mapping class groups contain elements admitting such factorizations. Moreover, for every pair of positive integers g,n, we tell whe…
Analysis of Hopf solitons as generalized fold maps
Hokkaido University（北海道大学）；Hiroshima University（广岛大学）；University of Colorado, Boulder（科罗拉多大学博尔德分校）；National Renewable Energy Laboratory（美国国家可再生能源实验室）；Keio University（庆应义塾大学）
