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.
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.
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.
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…
Floer theory relates the dynamics of Hamiltonian isotopies and the homology of the ambient manifold. It was extended to similarly relate the dynamics of symplectic isotopies and the Novikov homology associated to their flux. We discuss this picture regarding the fundamental group, and prove that when the flux is not too big, the associated Novikov fundamental group is generated by Floer moduli spaces associated to closed orbits of the symplectic isotopy.
Let $D$ be a 2-dimensional closed unit disk and $\rm{Symp}(D,0)_{\rm{rel}}$ the group of symplectomorphisms preserving the origin and the boundary $\partial D$ pointwise. We consider the $\mathbb{R}$-valued flux homomorphism on $\rm{Symp}(D,0)_{\rm{rel}}$ and define the central $\mathbb{R}$-extension called the $\mathbb{R}$-valued flux extension. We determine the Euler class of this extension and investigate the relation between the extension, the group $2$-cocycle defined by Ismagilov, Losik, and Michor, and the Calabi invariant of $D$.
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…
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 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.…
Given a smooth spacelike surface $Σ$ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation $ρ:π_1(S)\to\mathrm{PSL}_2\mathbb{R}\times\mathrm{PSL}_2\mathbb{R}$ where $S$ is a closed oriented surface of genus $\geq 2$, a canonical construction associates to $Σ$ a diffeomorphism $ϕ_Σ$ of $S$. It turns out that $ϕ_Σ$ is a symplectomorphism for the area forms of the two hyperbolic metrics $h$ and $h'$ on $S$ induced by the action of $ρ$ on $\mathbb{H}^2\times\mathbb{H}^2$. Using an algebraic construction related to the flux homomorphism, we give a new proof of …
Let $\hat{G}$ be a group and $G$ its normal subgroup. In this paper, we study $\hat{G}$-invariant quasimorphisms on $G$ which appear in symplectic geometry and low dimensional topology. As its application, we prove the non-existence of a section of the flux homomorphism on closed surfaces of higher genus. We also prove that Py's Calabi quasimorphism and Entov-Polterovich's partial Calabi quasimorphism are non-extendable to the group of symplectomorphisms. We show that Py's Calabi quasimorphism is the unique non-extendable quasimorphism to some group.
This paper meticulously revisit and study the flux geometry of any compact oriented manifold $(M; W)$. We generalize several well-known factorization results, exhibit some orbital conditions for the study of flux geometry, give a proof of the discreteness of the flux group for volume-preserving diffeomorphisms, show that any smooth path in the kernel of the flux is a vanishing flux path, and show that the kernel of the flux for volume-preserving diffeomorphisms is $C^1$closed inside the group of all volume-preserving diffeomorphisms isotopic to the identity map:This recovers several results fr…
The study of algebraic properties of groups of transformations of a manifold gives rise to an interplay between different areas of mathemathics such as topology, geometry, and dynamical systems. Especially, in this paper, we point out some interplays between topology, geometry, and dynamical systems which are underlying to the group of symplectic homeomorphisms. The latter situation can occur when one thinks of the following question. Is there a flux geometry which is underlying to the group of strong symplectic homeomorphisms so that Fathi's Poincare duality theorem continues to hold? We disc…
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…
In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented $3$-manifold $Y$ via the symplectic geometry of so-called traceless $\mathrm{SU}(2)$-character varieties. This invariant, $\mathrm{SI}(Y)$, which we refer to as the symplectic instanton homology of $Y$, was also shown to satisfy an exact triangle for Dehn surgeries on knots which is typical of Floer-theoretic invariants of $3$-manifolds. In this article, we demonstrate further structural properties of this symplectic instanton homology. For example, Floer theories are expected to roughly sati…
It is known that any Poisson manifold can be embedded into a bigger space which admites a description in terms of the canonical Poisson structure, i.e., Darboux coordinates. Such a procedure is known as a symplectic realization and has a number of important applications like quantization of the original Poisson manifold. In the present paper we extend the above idea to the case of quasi-Poisson structures which should not necessarily satisfy the Jacobi identity. For any given quasi-Poisson structure $Θ$ we provide a recursive procedure of the construction of a symplectic manifold, as well as t…
We construct a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold.
