The action--Maslov homomorphism $I\coπ_1(\text{Ham}(X,ω))\to\R$ is an important tool for understanding the topology of the Hamiltonian group of monotone symplectic manifolds. We explore conditions for the vanishing of this homomorphism, and show that it is identically zero when the Seidel element has finite order and the homology satisfies property $\mathcal{D}$ (a generalization of having homology generated by divisor classes). We use these results to show that $I=0$ for products of projective spaces and the Grassmannian of $2$ planes in $\C^4$.
We propose a topological theory of the Maslov index for lagrangian and symplectic paths based on a minimal system of axioms. We recover, as particular cases, the Hoermander and the Robbin-Salomon indices.
We revisit the definition of the Maslov index of loops in coisotropic submanifolds tangent to the characteristic foliation of this submanifold. This Maslov index is given by the mean index of a certain symplectic path which is a lift of the holonomy along the loop. We prove a result on the rigidity of the Maslov index for stable coisotropic submanifolds in a broad class of ambient symplectic manifolds. Furthermore, we establish a nearby existence theorem for the same class of ambient manifolds.
In this paper, we prove a splitting formula for the Maslov-type indices of symplectic paths induced by the splitting of the nullity in weak symplectic Hilbert space. Then we give a direct proof of iteration formulae for the Maslov-type indices of symplectic paths.
In this paper we find that the asymptotic nonlinear Maslov index defined on the universal cover of the group of all contact Hamiltonian diffeomorphisms of the standard 2n-1 dimensional contact sphere is a quasimorphism. Then we show our main result: Let M be standard n-1 complex projective space. We prove that the value of the pull back of the asymptotic nonlinear Maslov index to the universal cover of the group of Hamiltonian diffeomorphisms of M, restricted to elements supported in sufficiently small open subsets of M, equals the Calabi invariant of these elements up to a multiplication by a…
We examine symplectic topological features of certain family of monotone Lagrangian submanifolds in CP^n. Firstly, we give a cohomological restriction for Lagrangian submanifolds in CP^n whose first integral homologies are 3-torsion. In particular, in the case where n=5,8, we prove the cohomologies with coefficients in Z_2 of such Lagrangian submanifolds are isomorphic to that of SU(3)/(SO(3) Z_3) and SU(3)/Z_3, respectively. Secondly, we calculate the Floer cohomology of a monotone Lagrangian submanifold SU(p)/Z_p in CP^{p^2-1} with coefficients in Z_2 by using Biran-Cornea's theory.
Motivated by Mather theory of minimizing measures for symplectic twist dynamics, we study conformally symplectic flows on a cotangent bundle. These dynamics are the most general dynamics for which it makes sense to look at (asymptotic) dynamical Maslov index. Our main result is the existence of invariant measures with vanishing index without any convexity hypothesis, in the general framework of conformally symplectic flows. A degenerate twist-condition hypothesis implies the existence of ergodic invariant measures with zero dynamical Maslov index and thus the existence of points with zero dyna…
In this paper we detect new restrictions on the Maslov class of displaceable Lagrangian submanifolds of symplectic manifolds which are symplectically aspherical. These restrictions are established using action selectors for Hamiltonian flows. In particular, we construct and utilize a new action selector for the flows of a special class of Hamiltonian functions which arises naturally in the study of Hamiltonian paths which minimize the Hofer length functional.
In this note we prove a simple relation between the mean curvature form, symplectic area, and the Maslov class of a Lagrangian immersion in a Kähler-Einstein manifold. An immediate consequence is that in Kähler-Einstein manifolds with positive scalar curvature, minimal Lagrangian immersions are monotone.
We define Maslov--type indices associated to contact and symplectic transformation groups. There are two such families of indices. The first class of indices are maps from the homotopy groups of the contactomorphism or symplectomorphism group to a quotient of the integers. These are based on a generalization of the Maslov index. The second class of indices are maps from the homotopy groups of the space of contact structures or the space of cohomologous symplectic forms to the homotopy groups of a simple homogeneous space. We provide a detailed construction and describe some properties of these…
We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we define the curve's Maslov index by symplectic reduction to the classical finite-dimensional case. We prove the transitivity of repeated symplectic reductions and obtain the invariance of the Maslov index under symplectic reduction, while recovering all the standard properties of the …
We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.
A Maslov cycle is a singular variety in the lagrangian grassmannian L(V) of a symplectic vector space V consisting of all lagrangian subspaces having nonzero intersection with a fixed one. Givental has shown that a Maslov cycle is a Legendre singularity, i.e. the projection of a smooth conic lagrangian submanifold S in the cotangent bundle of L(V). We show here that S is the wavefront set of a Fourier integral distribution which is "evaluation at 0 of the quantizations".
In this paper we define the universal Maslov class of a Bohr - Sommerfeld lagrangian embedding to a simply connected pseudo - Einstein symplectic manifold. This class is an invariant of hamiltonian (isodrastic) deformations of a given Bohr - Sommerfeld lagrangian submanifold.
We use a neck stretching argument for holomorphic curves to produce symplectic disks of small area and Maslov class with boundary on Lagrangian submanifolds of nonpositive curvature. Applications include the proof of Audin's conjecture on the Maslov class of Lagrangian tori in linear symplectic space, the construction of a new symplectic capacity, obstructions to Lagrangian embeddings into uniruled symplectic manifolds, a quantitative version of Arnold's chord conjecture, and estimates on the size of Weinstein neighbourhoods. The main technical ingredient is transversality for the relevant mod…
In this work we define a relative version of the flux homomorphism, introduced by Calabi in 1969, for a symplectic manifold. We use it to study (the universal cover of) the group of symplectomorphisms of a symplectic manifold leaving a Lagrangian submanifold invariant. We show that some quotients of this group are stable under symplectic reduction.
In this note it is shown that the Maslov Index for pairs of Lagrangian Paths as introduced by Cappell, Lee and Miller appears by parallel transporting elements of (a certain complex line-subbundle of) the symplectic spinorbundle over Euclidean space, when pulled back to an (embedded) Lagrangian submanifold $L$, along closed or non-closed paths therein. In especially, the CLM-Index mod 4 determines the holonomy group of this line bundle w.r.t. the Levi-Civita-connection on $L$, hence its vanishing mod 4 is equivalent to the existence of a trivializing parallel section. Moreover, it is shown tha…
The first purpose of this paper is to generalize the well-known Maslov indices of maps of open Riemann surfaces with boundary lying on Lagrangian submanifolds to maps with boundary lying on coisotropic submanifolds in symplectic manifolds. For this purpose, we first define the notion of {\it Maslov loops} of coisotropic Grassmanians and their indices. Then we introduce the notions of {\it transverse Maslov bundle} of coisotropic submanifolds, and {\it gradable coisotropic submanifolds}. We then define {\it graded coisotropic submanifolds} and the {\it coisotropic Maslov index} of the maps with…
Gutzwiller's famous semiclassical trace formula plays an important role in theoretical and experimental quantum mechanics with tremendous success. We review the physical derivation of this deep periodic orbit theory in terms of the phase space formulation with an view towards the Hamiltonian dynamical systems. The Maslov phase appearing in the trace formula is clarified by Meinrenken as Conley-Zhender index for periodic orbits of Hamiltonian systems. We also survey and compare various versions of Maslov indices to establish this fact. A refinement and improvement to Conley-Zehnder's index theo…
In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit a metric of negative sectional curvature. Such Lagrangian submanifolds exist in every symplectic manifold of dimension greater than six or equal to four. The proof utilizes the relations between closed geodesics on the Lagrangian, the periodic orbits of geometric Hamiltonian flow…
We discuss symplectic manifolds where, locally, the structure is that encountered in Lagrangian dynamics. Exemples and characteristic properties are given. Then, we refer to the computation of the Maslov classes of a Lagrangian submanifold. Finally, we indicate the generalization of this type of structures to Poisson manifolds.
Let $(M,ω)$ be a symplectic manifold, $N\subseteq M$ a coisotropic submanifold, and $Σ$ a compact oriented (real) surface. I define a natural Maslov index for each continuous map $u:Σ\to M$ that sends every connected component of $\partialΣ$ to some isotropic leaf of $N$. This index is real valued and generalizes the usual Lagrangian Maslov index. The idea is to use the linear holonomy of the isotropic foliation of $N$ to compensate for the loss of boundary data in the case codimension $N<\dim M/2$. The definition is based on the Salamon-Zehnder (mean) Maslov index of a path of linear symplect…
Will Merry computed Rabinowitz Floer homology above Mane's critical value in terms of loop space homology by establishing an Abbondandolo-Schwarz short exact sequence. The purpose of this article is to provide an alternative proof of Merry's result. We construct a continuation homomorphism for symplectic deformations which enables us to reduce the computation to the untwisted case. Our construction takes advantage of a special version of the isoperimetric inequality which above Mane's critical value holds true.
One generalizes the notion of Maslov class of lagrangian embeddings to symplectic vector spaces for the compact case. Topological and geometrical properties of the generalized class is discussed. Certain relationship with the minimality problem is established. Applications are presented.
Let $X$ be a one-connected and integral symplectic manifold. In this paper, we construct and study a two-cocycle and three-cocycle on the symplectomorphism group of $X$. In particular, by using these cocycles, we clarify the relationship between Weinstein's action homomorphism and the universal Dixmier-Douady class of flat symplectic fibrations.
In this article we address two issues. First, we explore to what extent the techniques of Piunikhin, Salamon and Schwarz in [PSS96] can be carried over to Lagrangian Floer homology. In [PSS96] an isomorphism between Hamiltonian Floer homology and the singular homology is established. In contrast, Lagrangian Floer homology is not isomorphic to the singular homology of the Lagrangian submanifold, in general. Depending on the minimal Maslov number, we construct for certain degrees two homomorphisms between Lagrangian Floer homology and singular homology. In degrees where both maps are defined we …
We build homogeneous quasi-morphisms on the universal cover of the contactomorphism group for certain prequantizations of monotone symplectic toric manifolds. This is done using Givental's nonlinear Maslov index and a contact reduction technique for quasi-morphisms. We show how these quasi-morphisms lead to a hierarchy of rigid subsets of contact manifolds. We also show that the nonlinear Maslov index has a vanishing property, which plays a key role in our proofs. Finally we present applications to orderability of contact manifolds and Sandon-type metrics on contactomorphism groups.
This is a survey about certain "almost homomorphisms" and "almost linear" functionals (called quasi-morphisms and quasi-states) in symplectic topology and their applications to Hamiltonian dynamics, functional-theoretic properties of Poisson brackets and algebraic and metric properties of symplectomorphism groups.
We prove that quasi-morphisms and quasi-states on a closed integral symplectic manifold descend under symplectic reduction to symplectic hyperplane sections. Along the way we show that quasi-morphisms that arise from spectral invariants are the Calabi homomorphism when restricted to Hamiltonians supported on stably displaceable sets.
Let $(\X,ω)$ be a compact symplectic orbifold. We define $π_1(Ham(\X, ω))$, the fundamental group of the 2-group of Hamiltonian diffeomorphisms of $(\X, ω)$, and construct a group homomorphism from $π_1(Ham(\X, ω))$ to the group $QH_{orb}^*(\X,Λ)^{\times}$ of multiplicatively invertible elements in the orbifold quantum cohomology ring of $(\X, ω)$. This extends the Seidel representation ([Se], [M]) to symplectic orbifolds.
We prove using symplectic field theory that if the suspension of a hyperbolic diffeomorphism of the two-torus Lagrangian embeds in a closed uniruled symplectic six-manifold, then its image contains the boundary of a symplectic disc with vanishing Maslov index. This prevents such a Lagrangian submanifold to be monotone, for instance the real locus of a smooth real Fano manifold. It also prevents any Sol manifold to be in the real locus of an orientable real Del Pezzo fibration over a curve, confirming an expectation of J. Kollár. Finally, it constraints Hamiltonian diffeomorphisms of uniruled s…
We study pseudoholomorphic curves in symplectic quotients as adiabatic limits of solutions of a system of nonlinear first order elliptic partial differential equations in the ambient symplectic manifold. The symplectic manifold carries a Hamiltonian group action. The equations involve the Cauchy-Riemann operator over a Riemann surface, twisted by a connection, and couple the curvature of the connection with the moment map. Our main theorem asserts that the genus zero invariants of Hamiltonian group actions defined by these equations are related to the genus zero Gromov--Witten invariants of th…
This paper extends the flux homomorphism to volume-preserving homeomorphisms. A surprising $(C^0, δ)-$rigidity result where the extended flux groups coincide with the standard flux group is proved. The introduced tools, which also include a Poincaré duality with Fathi's mass flow and a norm on the group of volume-preserving homeomorphisms, indicate a potential for new flexibility in the behavior of homeomorphisms. This flexibility could have implications for rigidity results in symplectic/cosymplectic geometry, particularly those concerning Lefschetz manifolds: Any finite energy symplectic hom…
We define an integer graded symplectic Floer cohomology and a Fintushel-Stern type spectral sequence which are new invariants for monotone Lagrangian sub-manifolds and exact isotopes. The Z-graded symplectic Floer cohomology is an integral lifting of the usual Z_Sigma(L)-graded Floer-Oh cohomology. We prove the Kunneth formula for the spectral sequence and an ring structure on it. The ring structure on the Z_Sigma(L)-graded Floer cohomology is induced from the ring structure of the cohomology of the Lagrangian sub-manifold via the spectral sequence. Using the Z-graded symplectic Floer cohomolo…
Consider a sequence of compactly supported Hamiltonian diffeomorphisms $ϕ_k$ of an exact symplectic manifold, all of which are "graphical" in the sense that their graphs are identified by a Darboux-Weinstein chart with the image of a one-form. We show by an elementary argument that if the $ϕ_k$ $C^0$-converge to the identity then their Calabi invariants converge to zero. This generalizes a result of Oh, in which the ambient manifold was the two-disk and an additional assumption was made on the Hamiltonians generating the $ϕ_k$. We discuss connections to the open problem of whether the Calabi h…
For any closed oriented surface F of genus at least three, we prove the existence of foliated F-bundles over surfaces such that the signatures of the total spaces are non-zero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism from the symplectomophism group of the fiber to its first real cohomology. This crossed homomorphism extends the flux homomorphism defined on the identity component of the symplectomorphism group.
We construct an embedding of the full braid group on $m+1$ strands $B_{m+1}$, $m \geq 1$, into the contact mapping class group of the contactization $Q \times S^1$ of the $A_m$-Milnor fiber $Q$. The construction uses the embedding of $B_{m+1}$ into the symplectic mapping class group of $Q$ due to Khovanov and Seidel, and a natural lifting homomorphism. In order to show that the composed homomorphism is still injective, we use a partially linearized variant of the Chekanov--Eliashberg dga for Legendrians which lie above one another in $Q \times \mathbb{R}$, reducing the proof to Floer homology.…
The subject of this dissertation is the Gysin homomorphism in equivariant cohomology for spaces with torus action. We consider spaces which are quotients of classical semisimple complex linear algebraic groups by a parabolic subgroup with the natural action of a maximal torus, the so-called partial flag varieties. We derive formulas for the Gysin homomorphism for the projection to a point, of the form \[\int_X α= Res_{\mathbf{z}=\infty} \mathcal{Z}(\mathbf{z}, \mathbf{t}) \cdot α(\mathbf{t}),\] for a certain residue operation and a map $\mathcal{Z}(\mathbf{z}, \mathbf{t})$, associated to the r…
We extend the definition of Weinstein's Action homomorphism to Hamiltonian actions with equivariant moment maps of (possibly infinite-dimensional) Lie groups on symplectic manifolds, and show that under conditions including a uniform bound on the symplectic areas of geodesic triangles the resulting homomorphism extends to a quasimorphism on the universal cover of the group. We apply these principles to finite dimensional Hermitian Lie groups like Sp(2n,R), reinterpreting the Guichardet-Wigner quasimorphisms, and to the infinite dimensional groups of Hamiltonian diffeomorphisms Ham(M,\om) of cl…
We study the flux homomorphism for closed forms of arbitrary degree, with special emphasis on volume forms and on symplectic forms. The volume flux group is an invariant of the underlying manifold, whose non-vanishing implies that the manifold resembles one with a circle action with homologically essential orbits.
Consider a Maslov zero Lagrangian submanifold diffeomorphic to a Lie group on which an anti-symplectic involution acts by the inverse map of the group. We show that the Fukaya $A_\infty$ endomorphism algebra of such a Lagrangian is quasi-isomorphic to its de Rham cohomology tensored with the Novikov field. In particular, it is unobstructed, formal, and its Floer and de Rham cohomologies coincide. Our result implies that the smooth fibers of a large class of singular Lagrangian fibrations are unobstructed and their Floer and de Rham cohomologies coincide. This is a step in the SYZ and family Fl…
Let $(M,ω)$ be a closed symplectic manifold and $\textup{Ham}(M,ω)$ the group of Hamiltonian diffeomorphisms of $(M,ω)$. Then the Seidel homomorphism is a map from the fundamental group of $\textup{Ham}(M,ω)$ to the quantum homology ring $QH_*(M;Λ)$. Using this homomorphism we give a sufficient condition for when a nontrivial loop $ψ$ in $\textup{Ham}(M,ω)$ determines a nontrivial loop $ψ\times\textup{id}_N$ in $\textup{Ham}(M\times N,ω\oplusη)$, where $(N,η)$ is a closed symplectic manifold such that $π_2(N)=0$.
In this work we construct Calabi quasi-morphisms on the universal cover of the group Ham(M) of Hamiltonian diffeomorphisms for some non-monotone symplectic manifolds. This complements a result by Entov and Polterovich which applies in the monotone case. Moreover, in contrast to their work, we show that these quasi-morphisms descend to non-trivial homomorphisms on the fundamental group of Ham(M).
In this paper we find connection between the Hofer's metric of the group of Hamiltonian diffeomorphisms of a closed symplectic manifold, with an integral symplectic form, and the geometry, defined in a paper by Eliashberg and Polterovich, of the quantomorphisms group of its prequantization manifold. This gives two main results: First, we calculate, partly, the geometry of the quantomorphisms groups of a prequantization manifolds of an integral symplectic manifold which admits certain Lagrangian foliation. Second, for every prequantization manifold we give a formula for the distance between a p…
We further study the symplectic Khovanov homology of Seidel and Smith and its generalization to even tangles. This homology theory is a conjectural geometric model for Khovanov homology. In this paper we uncover structures on symplectic Khovanov homology which have analogues in Khovanov homology. To each elementary (as well as minimal) cobordism between two tangles we associate a homomorphism between the symplectic Khovanov homology groups of the two tangles. We define the symplectic analogues $H_{s}^m$ of Khovanov's arc algebras and equip the symplectic Khovanov homology of an $(m,n)$-tangle …
Symplectic Khovanov homology is an invariant of oriented links defined by Seidel and Smith and conjectured to be isomorphic to Khovanov homology. I define morphisms (up to a global sign ambiguity) between symplectic Khovanov homology groups, corresponding to isotopy classes of smooth link cobordisms in 4D between a fixed pair of links. These morphisms define a functor from the category of links and such cobordisms to the category of abelian groups and group homomorphisms up to a sign ambiguity. This provides an extra structure for symplectic Khovanov homology and more generally an isotopy inva…
We consider the action of symplectic monodromy on chain-level enhancements of quantum cohomology. First, we construct a family version of $A_\infty$-structure on quantum cohomology (this should morally correspond to Hochschild cohomology of a "family of Fukaya categories over the circle"). Following Kaledin, we look at the obstruction class of this structure, and argue that it can be related to a quantum version of Massey products on the one hand (which, in nice cases, can be related to actual counts of rational curves) and to the classical Andreadakis-Johnson theory of the Torelli group on th…
$A_\infty$ categories are a mathematical structure that appears in topological field theory, string topology, and symplectic topology. This paper studies the cyclic homology of a Calabi-Yau $A_\infty$ category, and shows that it is naturally an equivariant topological conformal field theory, and in particular, contains an involutive Lie bialgebra. Applications of the theory to string topology and the Fukaya category are given; in particular, it is shown that there is a Lie bialgebra homomorphism from the cyclic cohomology of the Fukaya category of a symplectic manifold with contact type bounda…
