Symplectic Fillings Research Articles

On symplectic fillings of small Seifert 3–manifolds

On symplectic fillings of small Seifert 3–manifolds

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 a smooth embedding of connected sums of lens spaces in C2 can be upgraded to a Stein embedding, and we have determined that this never happens.

Rational blowdown graphs for symplectic fillings of lens spaces

Rational blowdown graphs for symplectic fillings of lens spaces

On the cohomology ring of symplectic fillings

We consider symplectic cohomology twisted by sphere bundles, which can be viewed as an analogue of local systems. Using the associated Gysin exact sequence, we prove the uniqueness of part of the ring structure on cohomology of fillings for those asymptotically dynamically convex manifolds with vanishing property considered in [30,31]. In particular, for simply connected $4n+1$ dimensional flexible fillable contact $Y$, we show that real cohomology $H^*(W)$ is unique as a ring for any Liouville filling $W$ of $Y$ as long as $c_1(W)=0$. Uniqueness of real homotopy type of Liouville fillings is also obtained for a class of flexibly fillable contact manifolds.

Some applications of Menke’s JSJ decomposition for symplectic fillings

We apply Menke’s JSJ decomposition for symplectic fillings to several families of contact 3-manifolds. Among other results, we complete the classification up to orientation-preserving diffeomorphism of strong symplectic fillings of lens spaces. We show that exact symplectic fillings of contact manifolds obtained by surgery on certain Legendrian negative cables are the result of attaching a Weinstein 2-handle to an exact filling of a lens space. For large families of contact structures on Seifert fibered spaces over S 2 S^2 , we reduce the problem of classifying exact symplectic fillings to the same problem for universally tight or canonical contact structures. Finally, virtually overtwisted circle bundles over surfaces with genus greater than one and negative twisting number are seen to have unique exact fillings.

On uniqueness of symplectic fillings of links of some surface singularities

On uniqueness of symplectic fillings of links of some surface singularities

Bourgeois contact structures: Tightness, fillability and applications

Given a contact structure on a manifold V together with a supporting open book decomposition, Bourgeois gave an explicit construction of a contact structure on V times {mathbb {T}}^2. We prove that all such structures are universally tight in dimension 5, independent of whether the original contact manifold is itself tight or overtwisted. In arbitrary dimensions, we provide obstructions to the existence of strong symplectic fillings of Bourgeois manifolds. This gives a broad class of new examples of weakly but not strongly fillable contact 5-manifolds, as well as the first examples of weakly but not strongly fillable contact structures in all odd dimensions. These obstructions are particular instances of more general obstructions for {mathbb {S}}^1-invariant contact manifolds. We also obtain a classification result in arbitrary dimensions, namely that the unit cotangent bundle of the n-torus has a unique symplectically aspherical strong filling up to diffeomorphism.

Symplectic fillings of asymptotically dynamically convex manifolds II–k-dilations

We introduce the concept of k-(semi)-dilation for Liouville domains, which is a generalization of symplectic dilation defined by Seidel-Solomon. We prove that the existence of k-(semi)-dilation is a property independent of certain fillings for asymptotically dynamically convex (ADC) manifolds. We construct examples with k-dilations, but not k−1-dilations for all k⩾0. We extract invariants taking value in N∪{∞} for Liouville domains and ADC contact manifolds, which are called the order of (semi)-dilation. The order of (semi)-dilation serves as embedding and cobordism obstructions. We determine the order of (semi)-dilation for many Brieskorn varieties and use them to study cobordisms between Brieskorn manifolds.

Non‐simply connected symplectic fillings of lens spaces

We prove results exploring the relationship between the fundamental group and the second Betti number of minimal symplectic fillings of lens spaces. These results unify and generalize several disparate facts appearing in the literature. The Fibonacci numbers make a cameo appearance.

Seiberg–Witten Floer Homotopy Contact Invariant

We introduce a Floer homotopy version of the contact invariant introduced by Kronheimer–Mrowka–Ozsváth–Szabó. Moreover, we prove a gluing formula relating our invariant with the first author’s Bauer–Furuta type invariant, which refines Kronheimer–Mrowka’s invariant for 4-manifolds with contact boundary. As an application, we give a constraint for a certain class of symplectic fillings using equivariant KO-cohomology.

Symplectic fillings and cobordisms of lens spaces

We complete the classification of symplectic fillings of tight contact structures on lens spaces. In particular, we show that any symplectic filling X X of a virtually overtwisted contact structure on L ( p , q ) L(p,q) has another symplectic structure that fills the universally tight contact structure on L ( p , q ) L(p,q) . Moreover, we show that the Stein filling of L ( p , q ) L(p,q) with maximal second homology is given by the plumbing of disk bundles. We also consider the question of constructing symplectic cobordisms between lens spaces and report some partial results.

Surface singularities and planar contact structures

Dans cet article on démontre que si une structure de contact sur une variété de dimension trois est portée par un livre ouvert à pages planaires, alors une certaine configuration d’intersections n’apparaît dans l’homologie d’aucun de ses remplissages minimaux. On démontre de plus que les remplissages d’une telle variété de contact ne contiennent pas de surface symplectique de genre positif. En appliquant ces obstructions aux structures de contact canoniques sur les bords des singularités normales de surfaces, on montre que les bords des singularités isolées de surfaces dans l’espace complexe de dimension trois sont planaires seulement pour les singularités de type A n . En général, nous caractérisons complètement les bords planaires des singularités normales de surfaces (par leurs graphes de résolution) : ces singularités sont précisément les singularités rationnelles avec cycle fondamental réduit (aussi appelées singularités minimales). On montre aussi la non-planarité des structures de contact tendues sur certains petits L-espaces de Seifert ainsi que celle des structures de contact obtenues par la construction de Boothby–Wang appliquée aux surfaces de genre positif. De plus, on démontre que tout groupe de présentation finie est le groupe fondamental d’une fibration de Lefschetz à fibres planaires.

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 fillings yield the same torus bundle filling.

Symplectic, Poisson, and contact geometry on scattering manifolds

We introduce scattering-symplectic manifolds, manifolds with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. This paper will demonstrate the potential of the scattering symplectic setting. In particular, we construct scattering-symplectic spheres and scattering symplectic gluings between strong convex symplectic fillings of a contact manifold. By giving an explicit computation of the Poisson cohomology of a scattering symplectic manifold, we also introduce a new method of computing Poisson cohomology.

Symplectic fillings of asymptotically dynamically convex manifolds I

We consider exact fillings with vanishing first Chern class of asymptotically dynamically convex (ADC) manifolds. We construct two structure maps on the positive symplectic cohomology and prove that they are independent of the filling for ADC manifolds. The invariance of the structure maps implies that the vanishing of symplectic cohomology and the existence of symplectic dilations are properties independent of the filling for ADC manifolds. Using them, various topological applications on symplectic fillings are obtained, including the uniqueness of diffeomorphism types of fillings for many contact manifolds. We use the structure maps to define the first symplectic obstructions to Weinstein fillability. In particular, we show that for all dimension $4k+3, k\ge 1$, there exist infinitely many contact manifolds that are exactly fillable, almost Weinstein fillable but not Weinstein fillable. The invariance of the structure maps generalizes to strong fillings with vanishing first Chern class. We show that any strong filling with vanishing first Chern class of a class of manifolds, including $(S^{2n-1},\xi_{std}), \partial(T^*L \times \mathbb{C}^n)$ with $L$ simply connected, must be exact and have unique diffeomorphism type. As an application of the proof, we show that the existence of symplectic dilation implies uniruledness. In particular any affine exotic $\mathbb{C}^n$ with non-negative log Kodaira dimension is a symplectic exotic $\mathbb{C}^{n}$.

Disjoinable Lagrangian tori and semisimple symplectic cohomology

We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to certain birational surgeries like blow-downs and flips. As a consequence, there are many non-toric (non-compact) monotone symplectic manifolds whose wrapped Fukaya categories are proper.

Spine Removal Surgery and the Geography of Symplectic Fillings

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 [LVWa].

A Lefschetz fibration on minimal symplectic fillings of a quotient surface singularity

In this article, we construct a genus-0 or genus-1 positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a sequence of rational blowdowns from its minimal resolution.

Fillings and fittings of unit cotangent bundles of odd-dimensional spheres

AbstractWe introduce the concept of fittings to symplectic fillings of the unit cotangent bundle of odd-dimensional spheres. Assuming symplectic asphericity we show that all fittings are diffeomorphic to the respective unit co-disc bundle.

Symplectic Fillings, Contact Surgeries, and Lagrangian Disks

AbstractThis paper completely answers the question of when contact $(r)$–surgery on a Legendrian knot in the standard contact structure on $S^3$ yields a symplectically fillable contact manifold for $r\in (0,1]$. We also give obstructions for other positive $r$ and investigate Lagrangian fillings of Legendrian knots.