We construct a stable homotopy type invariant for any Legendrian submanifold in a jet bundle equipped with a linear-at-infinity generating family. We show that this spectrum lifts the generating family homology groups. When the generating family extends to a generating family for an embedded Lagrangian filling, we lift the Seidel isomorphism to the spectrum level. As applications, we establish topological constraints on Lagrangian fillings arising from generating families, algebraic constraints on whether generating families admit fillings, and lower bounds on how many fiber dimensions are needed to construct a generating family for a Legendrian.
We construct an infinite family of non-positive open books with structures. Combined with a result of Wendl, this allows us to give a complete answer to a long-standing question about the mapping class group of a compact surface with boundary: namely, we conclude that the monoid of monodromies supporting Stein-fillable contact structures is equal to the monoid of positive monodromies if and only if the surface is planar.
We use the symplectic rational blow-up to study some Lagrangian pinwheels in symplectic rational manifolds. In particular, we determine which symplectic forms in the threefold blow-up of P^2 carry Lagrangian projective planes that can be made disjoint by a Hamiltonian isotopy. In addition, we show that such a disjunction is not possible in del Pezzo surfaces with Euler characteristic between 4 and 7. Finally, we determine which symplectic forms on S^2× S^2 carry a Lagrangian L_3,1 pinwheel, answering a question of J. Evans.
In this article, we extend the methods from arXiv:2011.06568, where the five dimensional analogue of the three dimensional finite energy foliations introduced by Hofer--Wysocki--Zehnder was identified, to the case where there the underlying (IP) contact $5$-fold admits a $6$-dimensional (IP) symplectic filling. We show that the filling induces a moduli space of pseudo-holomorphic curves which is itself a symplectic filling of the standard $3$-sphere, and hence symplectomorphic to the $4$-dimensional ball. We further show that whenever the contact form on the $5$-fold is strictly convex, then this moduli space is a strictly convex domain, so that the induced Reeb dynamics at the boundary is in particular dynamically convex. For the circular restricted three-body problem, this implies that whenever the spatial dynamics (near one of the heavy masses) is strictly convex, the holomorphic shadow of arXiv:2011.06568 is dynamically convex; this is shown to hold for near-integrable cases close to the Kepler problem, and with mass ratio either zero or sufficiently close to 1.
In this paper, we consider the cohomology rings of some multiple weight varieties of type A, that is, symplectic torus quotients for a direct product of several coadjoint orbits of the special unitary group. Under some specific assumptions, we prove the symplectic volumes of multiple weight varieties are equal to the volumes of flow polytopes. Using differential equations satisfied by the volume functions of flow polytopes, we give an explicit presentation of the cohomology ring of the multiple weight variety of special type.
Let $K_0$ and $K$ be knots in $\mathbb{R}^3$. Suppose that by a compactly supported Hamiltonian isotopy on $T^*\mathbb{R}^3$, the conormal bundle of $K_0$ is isotopic to a Lagrangian submanifold which intersects the zero section cleanly along $K$. In this paper, we prove some constraints on the pair of knot types of $K_0$ and $K$. One example is that if $K_0$ is the unknot, then $K$ is also the unknot. We also consider some cases where $K_0$ and $K$ have specific knot types, such as torus knots and connected sums of trefoil knots. The key step is finding a DGA map between the Chekanov-Eliashberg DGAs of the unit conormal bundles of knots. The main results are deduced from a relation between the augmentation varieties of $K_0$ and $K$ determined by these DGAs.
We show that tori in Engel 4-manifolds behave analogously to knots in contact 3-manifolds: Every torus with trivial normal bundle is isotopic to infinitely many distinct transverse tori, distinguished locally (and globally in the nullhomologous case) by their formal invariants. (Few examples of transverse tori were previously known.) We classify the formal invariants, which are richer than for transverse knots. We show that in an overtwisted Engel structure, up to homotopy through such structures, these invariants are a complete set of uniqueness obstructions, and every torus with trivial normal bundle can be made transverse realizing any combination of these invariants. Fixing Engel structures not known to be overtwisted, we explore the range of the primary invariants of given tori. A sample application is that many Engel manifolds admit infinitely many transverse homotopy classes of unknotted transverse tori such that each class contains infinitely many transverse isotopy classes.
. We prove an involutive analog of the dual knot surgery formula of Eftekhary and Hedden-Levine. We also compute a small model for the local equivalence class of the involutive dual knot complex.
Many of the existing results for closed Hamiltonian G-manifolds are based on the analysis of the corresponding Hamiltonian functions using Morse-Bott techniques. In general such methods fail for non-compact manifolds or for manifolds with boundary. In this article, we consider circle actions only on symplectic manifolds that have (convex) contact type boundary. In this situation we show that many of the key ideas of Morse-Bott theory still hold, allowing us to generalize several results from the closed setting. Among these, we show that in our situation any symplectic group action is always Hamiltonian, we show several results about the topology of the symplectic manifold and in particular about the connectedness of its boundary. We also show that after attaching cylindrical ends, a level set of the Hamiltonian of a circle action is either empty or connected. We concentrate mostly on circle actions, but we believe that with our methods many of the classical results can be generalized from closed symplectic manifolds to symplectic manifolds with contact type boundary.
The concept moment map plays a central role in the study of Hamiltonian actions of compact Lie groups K on symplectic manifolds. In this note, we propose a theory of moment maps coupled with an AdK-invariant convex function f on k & lowast;, the dual of Lie algebra of K, and study the structure of the stabilizer of the critical point off composing with the moment map. As an outcome, we are able to obtain a general Calabi-Matsushima decomposition based only on the convexity of f so that all existing Calabi-Matsushima type of decomposition theorems fall into this new framework. This work is motivated by the work of Donaldson [Don17] together with the goal of finding a natural interpretation of Tian-Zhu's Calabidecomposition for Kaher-Ricci solitons in [TZ02], which are examples of infinite dimensional version of our setting.
The spectral diameter of a symplectic ball is shown to be equal to its capacity; this result upgrades the known bound by a factor of two and yields a simple formula for the spectral diameter of a symplectic ellipsoid. We also study the relationship between the spectral diameter and packings by two balls.
Plumbing spaces have drawn significant attention among symplectic topologists due to their natural occurrence as examples of Weinstein manifolds. In our paper, we provide an explicit general formula for the wrapped Fukaya category of plumbings (with arbitrary grading structure) of cotangent bundles along any quiver, in terms of homotopy types of cotangent bundles. The resulting category is freely generated by finitely many morphisms with differentials. Our approach relies on "local-to-global" computations. Especially, we give a specific presentation of the wrapped Fukaya category of "plumbing sectors" that serve as local models for the singularities of Lagrangian skeletons of plumbing spaces. As corollaries, we explicitly describe the wrapped Fukaya category of plumbing spaces in dimension 4 and plumbings of T*Sn for n >= 3. We show that any Ginzburg dg algebra/category of a graded quiver without potential is equivalent to the wrapped Fukaya category of a plumbing of T*Sn (with the corresponding grading structure).
We give a classification of generic coadjoint orbits for the group of area-preserving diffeomorphisms of a closed non-orientable surface. This completes V. Arnold's program of studying invariants of incompressible fluids in 2D. As an auxiliary problem, we also classify simple Morse pseudo-functions on non-orientable surfaces up to area-preserving diffeomorphisms.
Let L_0,L_1,L_2 ⊂ M be exact Lagrangian spheres in a Liouville domain M with 2c_1(M)=0. If L_0,L_1,L_2 form an A_3-configuration, we show that ℒ(L_0) and ℒ(L_2) endowed with the Hofer metric contain quasi-isometric embeddings of (ℝ^∞, ·_∞), i.e. infinite-dimensional quasi-flats. A corollary of the proof presented here establishes that Ham_c(M) itself contains an infinite-dimensional quasi-flat. We also show that for a Dehn twist τ: M → M along L_1 the boundary depth of CF(τ^2ℓ(L_0), L') is unbounded in L' ∈ℒ(L_2) for any ℓ∈ℕ_0.
In this paper we explain how to construct the EBK spectrum from the marked action spectrum and derive a minimax formula for concave toric domains. In the special case of the billiard on the disk we show that while the action spectrum is algebraic the EBK spectrum has infinite transcendence degree under the assumption that Schanuel's conjecture is true.
We define a higher-dimensional analogue of symplectic Khovanov homology. Consider the standard Lefschetz fibration p: W -> D subset of C of a 2n-dimensional Milnor fiber of the A2 kappa-1 singularity. We represent a link by a kappa-strand braid, which is expressed as an element h of the symplectic mapping class group Symp(W, partial derivative W). We then apply the higher-dimensional Heegaard Floer homology machinery to the pair (a, h(a)), where a is a collection of kappa unstable manifolds of W which are Lagrangian spheres. We prove its invariance under arc slides and Markov stabilizations, which shows that it is a link invariant. This work constitutes part of the author's PhD thesis.
