This paper studies the extension of the Hofer metric and general Finsler metrics on Hamiltonian symplectomorphism group $Ham(M,ω)$ to the identity component $Symp_0(M,ω)$ of symplectomorphism group. In particular, we prove that the Hofer metric on $Ham(M,ω)$ does not extend to a bi-invariant metric on $Symp_0(M,ω)$ for many symplectic manifolds. We also show that for the torus $T^{2n}$ with the standard symplectic form $ω$, no Finsler metric on $Ham(T^{2n},ω)$ that satisfies a strong form of the invariance condition can extend to a bi-invariant metric on $Symp_0(T^{2n},ω)$. Another interesting result is that there exists no $C^1$-continuous bi-invariant metric on $Symp_0(T^{2n},ω)$.
In this note we introduce parameterized Gromov-Witten invariants for symplectic fiber bundles and study the topology of the symplectomorphism group. We also give sample applications showing the non-triviality of certain homotopy groups of some symplectomorphism groups.
Consider any symplectic ruled surface $(M^g_λ,ω_λ)$ given by $(Σ_g \times S^2, λσ_{Σ_g} \oplus σ_{S^2})$. We compute all natural equivariant Gromov-Witten invariants $EGW_{g,0}(M^g_λ;H_k, A-kF)$ for all hamiltonian circle actions $H_k$ on $M^g_λ$, where $A=[Σ_g \times pt]$ and $F= [pt \times S^2]$. We use these invariants to show the nontriviality of certain higher order Whitehead products that live in the homotopy groups of the symplectomorphism groups $G_λ^g$, $g \geq 0$. Our results are sharper when $g=0,1$ and enable us to answer a question posed by D.McDuff in the case $g=1$ and provide a…
We consider an entropy-type invariant which measures the polynomial volume growth of submanifolds under the iterates of a map, and we establish sharp uniform lower bounds of this invariant for the following classes of symplectomorphisms of cotangent bundles over a compact base: 1) non-identical compactly supported symplectomorphisms which are symplectically isotopic to the identity, 2) symplectomorphisms generated by classical Hamiltonian functions, 3) Dehn twist like symplectomorphisms over compact rank one symmetric spaces.
Area and orientation preserving diffeomorphisms of the standard 2-disc, referred to as symplectomorphisms of $\mathbb{D}^{2}$, allow decompositions in terms of positive twist diffeomorphisms. Using the latter decomposition we utilize the Conley index theory of discrete braid classes as introduced in [Ghrist et al., C. R. Acad. Sci. Paris Sér. I Math., 331(11), 2000, Invent. Math., 152(2), 2003] in order to obtain a Morse type forcing theory of periodic points: a priori information about periodic points determines a mapping class which may force additional periodic points.
The special structures that arise in symplectic topology (particularly Gromov--Witten invariants and quantum homology) place as yet rather poorly understood restrictions on the topological properties of symplectomorphism groups. This article surveys some recent work by Abreu, Lalonde, McDuff, Polterovich and Seidel, concentrating particularly on the homotopy properties of the action of the group of Hamiltonian symplectomorphisms on the underlying manifold M. It sketches the proof that the evaluation map π_1(Ham(M))\to π_1(M) given by {ϕ_t}\mapsto {ϕ_t(x_0)} is trivial, as well as explaining si…
It has been shown that the Cauchy problem for geodesics in the space of Kähler metrics with a fixed cohomology class on a compact complex manifold $M$ can be effectively reduced to the problem of finding the flow of a related hamiltonian vector field $X_H$, followed by analytic continuation of the time to complex time. This opens the possibility of expressing the geodesic $ω_t$ in terms of Gröbner Lie series of the form $\exp(\sqrt{-1} \, tX_H)(f)$, for local holomorphic functions $f$. The main goal of this paper is to use truncated Lie series as a new way of constructing approximate solutions…
On groups of symplectomorphisms of the disk, we construct two homogeneous quasi-morphisms which relate to the Calabi invariant and the flux homomorphism respectively. We also show the relation between the quasi-morphisms and the translation number introduced by Poincaré.
We prove that the vector space R^d of any finite dimension d with the standard metric embeds in a bi-Lipschitz way into the group of area-preserving diffeomorphisms G of the two-sphere endowed with the L^p-metric for p>2. Along the way we show that the L^p-metric on the group G is unbounded for p>2 by elementary methods.
We study the symplectomorphism groups $G_λ=Symp_0(M,ω_λ)$ of an arbitrary closed manifold M equipped with a 1-parameter family of symplectic forms $ω_λ$ with variable cohomology class. We show that the existence of nontrivial elements in $π_*({\cal A},{\cal A}')$, where $({\cal A},{\cal A}')$ is a suitable pair of spaces of almost complex structures, implies the exiarxiv.org stence of families of nontrivial elements in $π_{*-i}G_λ$, for $i=1$ or 2. Suitable parametric Gromov Witten invariants detect nontrivial elements in $π_*({\cal A},{\cal A}')$. By looking at certain resolutions of quotient…
These notes combine material from short lecture courses given in Paris, France, in July 2001 and in Srni, the Czech Republic, in January 2003. They discuss groups of symplectomorphisms of closed symplectic manifolds (M,\om) from various points of view. Lectures 1 and 2 provide an overview of our current knowledge of their algebraic, geometric and homotopy theoretic properties. Lecture 3 sketches the arguments used by Gromov, Abreu and Abreu-McDuff to figure out the rational homotopy type of these groups in the cases M= CP^2 and M=S^2\times S^2. We outline the needed J-holomorphic curve techniq…
The spectral metric, defined by Schwarz and Oh using Floer-theoretical method, is a bi-invariant metric on the Hamiltonian diffeomorphism group. We show in this note that for certain symplectic manifolds, this metric can not be extended to a bi-invariant metric on the full group of symplectomorphisms. We also study the bounded isometry conjecture of Lalonde and Polterovich in the context of the spectral metric. In particular, we show that the conjecture holds for the torus with all linear symplectic forms.
We introduce here a natural functional associated to any $b \in QH_* (M, ω)$: \emph{spectral length functional}, on the space of "generalized paths" in $ \text {Ham}(M, ω)$, closely related to both the Hofer length functional and spectral invariants and establish some of its properties. This functional is smooth on its domain of definition, and moreover the nature of extremals of this functional suggests that it may be variationally complete, in the sense that any suitably generic element of $ \widetilde{\text {Ham}}(M, ω)$ is connected to $id$ by a generalized path minimizing spectral length.…
Let $X$ be any rational ruled symplectic four-manifold. Given a symplectic embedding $ι:B_{c}\into X$ of the standard ball of capacity $c$ into $X$, consider the corresponding symplectic blow-up $\tX_ι$. In this paper, we study the homotopy type of the symplectomorphism group $\Symp(\tX_ι)$, simplifying and extending the results of math.SG/0207096. This allows us to compute the rational homotopy groups of the space $\IEmb(B_{c},X)$ of unparametrized symplectic embeddings of $B_{c}$ into $X$. We also show that the embedding space of one ball in $CP^2$, and the embedding space of two disjoint ba…
This paper concerns the action of linear symplectomorphisms on linear symplectic forms by conjugation in even dimensions. We prove that pfaffian and $-\frac{1}{2}\operatorname{tr}(JA)$ (sum function) of $A$ are invariants on the action. We use these invariants to provide a complete description of the orbit space in dimension four. In addition, we investigate the geometric shapes of the individual orbits in dimension four. In symplectic geometry, our classification result in dimension four provides a necessary condition for two symplectic forms on $\mathbb{R}^{4}$ to be intertwined by symplecto…
We show that a minimal surface of general type has a canonical symplectic structure (unique up to symplectomorphism) which is invariant for smooth deformation. We show that the symplectomorphism type is also invariant for deformations which allow certain normal singularities, provided one remains in the same smoothing component. We use this technique to show that the Manetti surfaces yield examples of surfaces of general type which are not deformation equivalent but are canonically symplectomorphic.
The main theme of this paper is to study for a symplectomorphism of a compact surface, the asymptotic invariant which is defined to be the growth rate of the sequence of the total dimensions of symplectic Floer homologies of the iterates of the symplectomorphism. We prove that the asymptotic invariant coincides with asymptotic Nielsen number and with asymptotic absolute Lefschetz number. We also show that the asymptotic invariant coincides with the largest dilatation of the pseudo-Anosov components of the symplectomorphism and its logarithm coincides with the topological entropy. This implies …
We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp \TM and Diff \TM of its one point blow up \TM. There are three main arguments. The first shows that for any oriented M the natural map from pi_1(M) to pi_0(Diff \TM) is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group pi_1(Diff \TM) that persist into the space of self homotopy equivalences of \TM. Since it uses purely homological arguments, it applies to …
We use the criteria of Lalonde and McDuff to show that a path that is generated by a generic autonomous Hamiltonian is length minimizing with respect to the Hofer norm among all homotopic paths provided that it induces no non-constant closed trajectories in M. This generalizes a result of Hofer for symplectomorphisms of Euclidean space. The proof for general M uses Liu-Tian's construction of S^1-invariant virtual moduli cycles. As a corollary, we find that any semifree action of S^1 on M gives rise to a nontrivial element in the fundamental group of the symplectomorphism group of M. We also es…
In his 1992 article on generating functions Viterbo constructed a bi-invariant metric on the group of compactly supported Hamiltonian symplectomorphisms of R^2n. Using the set-up of arXiv:0901.3112 we extend the Viterbo metric to the group of compactly supported contactomorphisms of R^2n x S^1 isotopic to the identity. We also prove that the contactomorphism group is unbounded with respect to this metric.
We give a classification of generic coadjoint orbits for the groups of symplectomorphisms and Hamiltonian diffeomorphisms of a closed symplectic surface. We also classify simple Morse functions on symplectic surfaces with respect to actions of those groups. This gives an answer to V.Arnold's problem on describing all invariants of generic isovorticed fields for the 2D ideal fluids. For this we introduce a notion of anti-derivatives on a measured Reeb graph and describe their properties.
Elements of the tropical vertex group, introduced by Kontsevich and Soibelman, are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are equivalent to calculations of certain genus 0 relative Gromov-Witten invariants of toric surfaces. The relative invariants which arise have full tangency to a toric divisor at a single unspecified point. The method uses scattering diagrams, tropical curve counts, degeneration formulas, and exact multiple cover calculations in orbifold Gromov-Witten theory.
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.
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 $D$.
In this paper, we construct infinitely many bi-invariant metrics on the Hamiltonian diffeomorphism group and study their basic properties and corresponding generalizations of the Hofer inequality and Sikorav one.
We prove that bi-invariant word metrics are bounded on certain Chevalley groups. As an application we provide restrictions on Hamiltonian actions of such groups.
