Research

Abstract: We prove that Bourgeois' contact structures on $M\times T^{2}$ determined by a supporting open book for a contact manifold $(M,\xi)$ are tight. The proof is based on a contact homology computation leveraging holomorphic foliations and Kuranishi structures.

Abstract: We execute Avdek's algorithm to find many algebraically overtwisted and tight 3-manifolds by contact +1 surgeries. In particular, we show that a contact 1/k surgery on the standard contact 3-sphere  along any positive torus knot with the maximum Thurston–Bennequin invariant yields an algebraically overtwisted and tight 3-manifold, where k is a positive integer.

Abstract: We extend the hierarchy functors of [30] for the case of strong symplectic cobordisms, via deformations with Maurer--Cartan elements. In particular, we prove that the concave boundary of a strong cobordism has finite algebraic planar torsion if the convex boundary does, which yields a functorial proof of finite algebraic planar torsion for contact manifolds admitting strong cobordisms to overtwisted contact manifolds. We also show the existence of contact 3-folds without strong cobordisms to the standard contact 3-sphere, that are not cofillable. We also include generalizations of the theory, which appeared in an earlier version of [30], relating our notion of algebraic planar torsion to Latschev--Wendl's notion of algebraic torsion, discussing variations from counting holomorphic curves with general constraints and invariants extracted from higher genera holomorphic curves from an algebraic perspective.

Abstract: We show that a contact (+1)-surgery along a Legendrian sphere in a flexibly fillable contact manifold (c_1=0 if not subcritical) yields a contact manifold that is algebraically overtwisted if the Legendrian's homology class is not annihilated in the filling. Our construction can also be implemented in more general contact manifolds yielding algebraically overtwisted manifolds through (+1)-surgeries. This gives new proof of the vanishing of contact homology for overtwisted contact manifolds. Our result can be viewed as the symplectic field theory analog in any dimension of the vanishing of contact Ozsváth-Szabó invariant for (+1)-surgeries on two-component Legendrian links proved by Ding, Li, and Wu.

Abstract: We use spinal open books to construct contact manifolds with infinitely many different Weinstein fillings in any odd dimension > 1, which were previously unknown for dimensions equal to 4n+1. The argument does not involve understanding factorizations in the symplectic mapping class group.

Abstract: For any n > 1, we prove that S^{2n+1} admits a tight non-fillable contact structure that is homotopically standard. By taking connected sums we deduce that, for n > 2, any 2n+1-dimensional manifold that admits a tight contact structure, also admits a tight but not strongly fillable contact structure, in the same almost contact class. For n > 2, we further construct infinitely many Liouville but not Weinstein fillable contact structures on S^{2n+1} that are homotopically standard.

Abstract: We prove, by an ad hoc method, that exact fillings with vanishing rational first Chern class of flexibly fillable contact manifolds have unique integral intersection forms. We appeal to the special Reeb dynamics (stronger than ADC à la Lazarev) on the contact boundary, while a more systematic approach working for general ADC manifolds is developed independently by Eliashberg, Ganatra and Lazarev. We also discuss cases where the vanishing rational first Chern class assumption can be removed. We derive the uniqueness of diffeomorphism types of exact fillings of certain flexibly fillable contact manifolds and obstructions to contact embeddings, which are not necessarily exact.

Abstract: We study several aspects of fillings for links of general quotient singularities using Floer theory, including co-fillings, Weinstein fillings, strong fillings, exact fillings and exact orbifold fillings, focusing on non-existence of exact fillings of contact links of isolated terminal quotient singularities. We provide an extensive list of isolated terminal quotient singularities whose contact links are not exactly fillable, including \mathbb{C}^n/\mathbb{Z}/2) for n> 2, which settles a conjecture of Eliashberg, quotient singularities from general cyclic group actions and finite subgroups of SU(2), and all terminal quotient singularities in complex dimension 3. We also obtain uniqueness of the orbifold diffeomorphism type of exact orbifold fillings of contact links of some isolated terminal quotient singularities.

Abstract: We extend Donaldson's asymptotically holomorphic techniques to symplectic orbifolds. More precisely, given a symplectic orbifold such that the symplectic form defines an integer cohomology class, we prove that there exist sections of large tensor powers of the prequantizable line bundle such that their zero sets are symplectic suborbifolds. We then derive a Lefschetz hyperplane theorem for these suborbifolds, that computes their real cohomology up to middle dimension. We also get the hard Lefschetz and formality properties for them, when the ambient manifold satisfies those properties. 

For n> 4, we show that there are infinitely many formally contact isotopic embeddings of the standard ST^*S^{n−1}to (S^{2n−1},\xi_{std}) that are not contact isotopic. This answers a conjecture of Casals and Etnyre except for the \(n=3\) case. The argument does not appeal to the surgery formulae of critical handle attachement for Floer theory/SFT. 

Abstract: We study exact orbifold fillings of contact manifolds using Floer theories. Motivated by Chen-Ruan's orbifold Gromov-Witten invariants, we define symplectic cohomology of an exact orbifold filling as a group using classical techniques, i.e. choosing generic almost complex structures. By studying moduli spaces of pseudo-holomorphic curves in orbifolds, we obtain various non-existence, restrictions and uniqueness results for orbifold singularities of exact orbifold fillings of many contact manifolds. For example, we show that exact orbifold fillings of (\mathbb{RP}^{2n-1},\xi_{\mathrm{std}}) always have exactly one singularity modeled on \mathbb{C}^n/(\mathbb{Z}/2\mathbb{Z}) if n\ne 2^k. Lastly, we show that in dimension at least 3 there are pairs of contact manifolds without exact cobordisms in either direction, and that the same holds for exact orbifold cobordisms in dimension at least 5. 

Abstract: We define a hierarchy functor from the exact symplectic cobordism category to a totally order set from a BL_\infty (Bi-Lie) formalism of the rational symplectic field theory (RSFT). The hierarchy functor consists of three levels of structures, namely algebraic planar torsion, order of semi-dilations and planarity, all taking values in \mathbb{N}\cup \{\infty\}, where algebraic planar torsion can be understood as the analogue of Latschev-Wendl's algebraic torsion in the context of RSFT. The hierarchy functor is well-defined through a partial construction of RSFT and is within the scope of established virtual techniques. We develop computation tools for those functors and prove all three of them are surjective. In particular, the planarity functor is surjective in all dimension \ge 3. Then we use the hierarchy functor to study the existence of exact cobordisms. We discuss examples including iterated planar open books, spinal open books, affine varieties with uniruled compactifications and links of singularities.

Abstract: We show that the minimal symplectic area of Lagrangian submanifolds are universally bounded in symplectically aspherical domains with vanishing symplectic cohomology. If an exact domain admits a k-semi-dilation, then the minimal symplectic area is universally bounded for K(\pi,1)-Lagrangians. As a corollary, we show that Arnold chord conjecture holds for the following four cases: (1) Y admits an exact filling with SH^*(W)=0; (2) Y admits a symplectically aspherical filling with SH^*(W)=0 and simply connected Legendrians; (3) Y admits an exact filling with a k-semi-dilation and the Legendrian is a K(\pi,1) space; (4) Y is the cosphere bundle S^*Q with \pi_2(Q)\to H_2(Q) nontrivial and the Legendrian has trivial \pi_2. In addition, we obtain the existence of homoclinic orbits in case (1). We also provide many more examples with k-semi-dilations in all dimensions \ge 4. 

Abstract:  We show that symplectically aspherical/Calabi-Yau filling of Y:=\partial(V\times \mathbb{D}) has vanishing symplectic cohomology for any Liouville domain V, in particular, we make no topological requirement on the filling and c_1(V) can be nonzero. Moreover, we show that for any symplectically aspherical/Calabi-Yau filling W of Y, the interior \mathring{W} is diffeomorphic to the interior of V\times \mathbb{D} if \pi_1(Y) is abelian and \dim V\ge 4. And W is diffeomorphic to V\times\mathbb{D} if moreover the Whitehead group of \pi_1(Y) is trivial.

Abstract: We prove that (\mathbb{RP}^{2n-1},\xi_{std}) is not exactly fillable for any n\ne 2^k and there exist strongly fillable but not exactly fillable contact manifolds for all dimension \ge 5. 

Abstract: We introduce the concept of k-(semi)-dilation for Liouville domains, which is a generalization of symplectic dilation defined by Seidel-Solomon. We prove the existence of k-(semi)-dilation is a property independent of fillings for asymptotically dynamically convex (ADC) manifolds. We construct examples with k-dilations, but not k−1-dilations for all k\ge 0. We extract invariants taking value in \mathbb{N}\cup\{\infty\} 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.

Abstract: We consider exact fillings with vanishing first Chern class of asymptotically dynamically convex (ADC) manifolds. We construct two structure maps on positive symplectic cohomology and prove that they are independent of the filling for ADC manifolds. The invariance of the structure maps implies that vanishing of symplectic cohomology and existence of symplectic dilation are properties independent of the filling for ADC manifolds. Using them, various topological applications on symplectic fillings are obtained, including the uniqueness of diffeomorphism type 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×\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. 

Abstract: In this paper, we construct cochain complexes generated by cohomology of critical manifolds for Morse-Bott theory under minimum transversality assumptions. We discuss the relations between different constructions of cochain complexes for Morse-Bott theory. In particular, we explain how homological perturbation theory is used in Morse-Bott cohomology, and both our construction and the cascade construction can be interpreted in that way. In the presence of group actions, we construct cochain complexes for the equivariant theory. Expected properties like the independence of approximation of the classifying spaces and existence of the action spectral sequence are proven. We carry out our construction for finite dimensional Morse-Bott cohomology using a generic metric and prove it recovers the regular cohomology. We outline the project of combining our construction with polyfold perturbation theory. 

Abstract: 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 [27,29]. 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. 

We introduce group actions on polyfolds and polyfold bundles. We prove quotient theorems for polyfolds, when the group action has finite isotropy. We prove that the sc-Fredholm property is preserved under quotient if the base polyfold is infinite dimensional. The quotient construction is the main technical tool in the construction of equivariant fundamental class in [42]. We also analyze the equivariant transversality near the fixed locus in the polyfold setting. In the case of S^1-action with fixed locus, we give a sufficient condition for the existence of equivariant transverse perturbations. We outline the application to Hamiltonian-Floer cohomology and a proof of the weak Arnold conjecture for general symplectic manifolds, assuming the existence of Hamiltonian-Floer cohomology polyfolds. 

Abstract: We construct counterexamples to classical calculus facts such as the Inverse and Implicit Function Theorems in Scale Calculus -- a generalization of Multivariable Calculus to infinite dimensional vector spaces in which the reparameterization maps relevant to Symplectic Geometry are smooth. Scale Calculus is a cornerstone of Polyfold Theory, which was introduced by Hofer-Wysocki-Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves. We show how the novel nonlinear scale-Fredholm notion in Polyfold Theory overcomes the lack of Implicit Function Theorems, by formally establishing an often implicitly used fact: The differentials of basic germs -- the local models for scale-Fredholm maps -- vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of Polyfold Theory. 

Abstract: For any asymptotically dynamically convex contact manifold Y, we show that SH^*(W)=0 is a property independent of the choice of topologically simple (i.e., c_1(W)=0 and \pi_1(Y)\to \pi_1(W) is injective) Liouville filling W⁠. In particular, if Y is the boundary of a flexible Weinstein domain, then any topologically simple Liouville filling W has vanishing symplectic homology. As a consequence, we answer a question of Lazarev partially: a contact manifold Y admitting flexible fillings determines the integral cohomology of all the topologically simple Liouville fillings of Y. The vanishing result provides an obstruction to flexible fillability. As an application, we show that all Brieskorn manifolds of dimension \ge 5 cannot be filled by flexible Weinstein manifolds. 

Abstract: In this paper, we propose a general method of defining equivariant theories in symplectic geometry using polyfolds. The construction is twofold, one is for closed theories like equivariant Gromov-Witten theory, the other is for open theories like equivariant Floer cohomology.

When a compact Lie group G acts on a tame strong polyfold bundle p:W \to Z, we construct a quotient polyfold bundle \overline{p}:W/G \to Z/G if the G-action on Z only has finite isotropy. For a general group action and if Z has no boundary, then every G-equivariant sc-Fredholm section s: Z \to W induces a H^*(BG) module map s_*: H^*_G(Z) \to H^{*-\ind s}(BG), which can be viewed as a generalization of the integration over the zero set s^{-1}(0) when equivariant transversality holds. When Z is the Gromov-Witten polyfold, s_* yields a definition equivariant Gromov-Witten invariant for any symplectic manifold. We obtain a localization theorem for s_* if there exist tubular neighborhoods around the fixed locus in the sense of polyfold. For open theories, we first obtain a construction for the Morse-Bott theories under minimal transversality requirement. We discuss the relations between different constructions of cochain complexes for Morse-Bott theory. We explain how homological perturbation theory is used in Morse-Bott cohomology, in particular, both our construction and the cascades construction can be interpreted in that way, In the presence of group actions, we construct cochain complexes for the equivariant theory. Expected properties like the independence of approximations of the classifying spaces and existence of action spectral sequences are proven. We carry out our construction for finite dimensional Morse-Bott cohomology using a generic metric and prove it recovers the regular cohomology. We outline the project of combining our construction with polyfold theory, which is expected to give a general construction for both Morse-Bott and equivariant Floer cohomology.

In the last part, we show that for any asymptotically dynamically convex contact manifold Y, the vanishing of symplectic homology SH(W)=0 is a property independent of the choice of topologically simple (i.e. c_1(W)=0 and \pi_{1}(Y)\to \pi_1(W) is injective) Liouville filling W. As a consequence, we answer a question of Lazarev partially: a contact manifold Y admitting flexible fillings determines the integral cohomology of all the topologically simple Liouville fillings of Y.

Abstract: For every equivariant Fredholm problem of strong polyfold bundle without boundary, we construct a H^*(BG) module homomorphism (the equivariant fundamental class) from the equivariant cohomology of the base polyfold H^∗_G(Z, \tau_i) to H^*(BG). It can be viewed as a generalization of integration over the zero set, when equivariant transversality is possible. We prove a localization theorem for equivariant fundamental class under some assumptions on the fixed locus.

Remark: Roughly speaking, the assumption on fixed locus is the existence of a global Kuranishi chart near the fixed locus. Such structure may not exist in the general setting considered in this paper. However, such global charts exist for Gromov-Witten theory by the work of Abouzaid, McLean and Smith. As AMS's construction gives the whole moduli space a global chart, localization can be considered on level of global charts instead of polyfolds, which is much simpler.