In this paper we discuss the topology of the symplectomorphism group of a product of two 2-dimensional spheres when the ratio of their areas lies in the interval (1,2]. More precisely we compute the homotopy type of this symplectomorphism group and we also show that the group contains two finite dimensional Lie groups generating the homotopy. A key step in this work is to calculate the mod 2 homology of the group of symplectomorphisms. Although this homology has a finite number of generators with respect to the Pontryagin product, it is unexpected large containing in particular a free noncommutative ring with 3 generators.
Let ${\cal O}$ be a quantizable coadjoint orbit of a semisimple Lie group $G$. Under certain hypotheses we prove that $#(π_1(\text{Ham}({\cal O})))\geq #(Z(G))$, where $\text{Ham}({\cal O})$ is the group of Hamiltonian symplectomorphisms of ${\cal O}$.
By a result of Kedra and Pinsonnault, we know that the topology of groups of symplectomorphisms of symplectic 4-manifolds is complicated in general. However, in all known (very specific) examples, the rational cohomology rings of symplectomorphism groups are finitely generated. In this paper, we compute the rational homotopy Lie algebra of symplectomorphism groups of the 3-point blow-up of the projective plane (with an arbitrary symplectic form) and show that in some cases, depending on the sizes of the blow-ups, it is infinite dimensional. Moreover, we explain how the topology is generated by…
We continue the analysis, started by Abreu, McDuff and Anjos, of the topology of the group of symplectomorphisms of $S^2 \times S^2$ when the ratio of the areas of the two spheres lies in the interval (1,2]. We express the group, up to homotopy, as the amalgam of certain of its compact Lie subgroups. We use this to compute the homotopy type of the classifying space of the group of symplectomorphisms and the corresponding ring of characteristic classes for symplectic fibrations.
We consider the 3-point blow-up of the manifold $ (S^2 \times S^2, σ\oplus σ)$ where $σ$ is the standard symplectic form which gives area 1 to the sphere $S^2$, and study its group of symplectomorphisms $\rm{Symp} ( S^2 \times S^2 \#\, 3\overline{\mathbb C\mathbb P}\,\!^2, ω)$. So far, the monotone case was studied by J. Evans and he proved that this group is contractible. Moreover, J. Li, T. J. Li and W. Wu showed that the group Symp$_{h}(S^2 \times S^2 \#\, 3\overline{ \mathbb C\mathbb P}\,\!^2,ω) $ of symplectomorphisms that act trivially on homology is always connected and recently they al…
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.
The symplectomorphism group of a 2-dimensional surface is homotopy equivalent to the orbit of a filling system of curves. We give a generalization of this statement to dimension 4. The filling system of curves is replaced by a decomposition of the symplectic 4-manifold (M, omega) into a disjoint union of an isotropic 2-complex L and a disc bundle over a symplectic surface Sigma which is Poincare dual to a multiple of the form omega. We show that then one can recover the homotopy type of the symplectomorphism group of M from the orbit of the pair (L, Sigma). This allows us to compute the homoto…
This paper studies groups of symplectomorphisms of ruled surfaces for symplectic forms with varying cohomology class. This class is characterized by the ratio R of the size of the base to that of the fiber. By considering appropriate spaces of almost complex structures, we investigate how the topological type of these groups changes as R increases. If the base is a sphere, this changes precisely when R passes an integer, and for general bases it stabilizes as R goes to infinity. Our results extend and make more precise some of the conclusions of Abreu--McDuff concerning the rational homotopy t…
We classify generic coadjoint orbits for symplectomorphism groups of compact symplectic surfaces with or without boundary. We also classify simple Morse functions on such surfaces up to a symplectomorphism.
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…
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…
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…
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.
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…
Starting from a given norm on the vector space of exact 1-forms of a compact symplectic manifold, we produce pseudo-distances on its symplectomorphism group by generalizing an idea due to Banyaga. We prove that in some cases (which include Banyaga's construction), their restriction to the Hamiltonian diffeomorphism group is equivalent to the distance induced by the initial norm on exact 1-forms. We also define genuine "distances to the Hamiltonian diffeomorphism group" which we use to derive several consequences, mainly in terms of flux groups.
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…
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 …
Let $M$ be either $S^2\times S^2$ or the one point blow-up $\cp# \bcp$ of $\cp$. In both cases $M$ carries a family of symplectic forms $\om_\la$, where $\la > -1$ determines the cohomology class $[\om_\la]$. This paper calculates the rational (co)homology of the group $G_\la$ of symplectomorphisms of $(M,\om_\la)$ as well as the rational homotopy type of its classifying space $BG_\la$. It turns out that each group $G_\la$ contains a finite collection $K_k, k = 0,...,\ell = \ell(\la)$, of finite dimensional Lie subgroups that generate its homotopy. We show that these subgroups "asymptotically …
On the space ${\cal L}$, of loops in the group of Hamiltonian symplectomorphisms of a symplectic quantizable manifold, we define a closed ${\bf Z}$-valued 1-form $Ω$. If $Ω$ vanishes, the prequantization map can be extended to a group representation. On ${\cal L}$ one can define an action integral as an ${\bf R}/{\bf Z}$-valued function, and the cohomology class $[Ω]$ is the obstruction to the lifting of that action integral to an ${\bf R}$-valued function. The form $Ω$ also defines a natural grading on $π_1({\cal L})$.
We show that, for certain families $ϕ_{\mathbf{s}}$ of diffeomorphisms of high-dimensional spheres, the commutator of the Dehn twist along the zero-section of $T^*S^n$ with the family of pullbacks $ϕ^*_{\mathbf{s}}$ gives a noncontractible family of compactly-supported symplectomorphisms. In particular, we find examples: where the Dehn twist along a parametrised Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation; where the symplectomorphism group is not simply-connected, and where the symplectomorphism group does not have the homotopy-type of a finite CW-complex. We sho…
In this survey article, we summarize some recent progress and problems on the symplectomorphism groups, with an emphasis on the connection to the space of ball-packings.
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…
For a symplectic manifold $(M,\om)$ with exact symplectic form we construct a 2-cocycle on the group of symplectomorphisms and indicate cases when this cocycle is not trivial.
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 construct some canonically defined central extensions of groups of symplectomorphisms. We show that this central extension is nontrivial in the case of a torus of dimension $\ge 6$ and in the case of a two-dimensional surface of genus $\ge 3$.
Let (M,ω) be a compact symplectic manifold, and ϕbe a symplectic diffeomorphism on M, we define a Floer-type homology FH_*(ϕ) which is a gen- eralization of Floer homology for symplectic fixed points defined by Dostoglou and Salamon for monotone symplectic manifolds. These homology groups are modules over a suitable Novikov ring and depend only on ϕup to a Hamiltonian isotopy.
Consider a compact Kähler manifold endowed with a prequantum bundle. Following the geometric quantization scheme, the associated quantum spaces are the spaces of holomorphic sections of the tensor powers of the prequantum bundle. In this paper we construct an asymptotic representation of the prequantum bundle automorphism group in these quantum spaces. We estimate the characters of these representations under some transversality assumption. The formula obtained generalizes in some sense the Lefschetz fixed point formula for the automorphisms of the prequantum bundle preserving its holomorphic …
We continue our previous work to prove that for any non-minimal ruled surface $(M,ω)$, the stability under symplectic deformations of $π_0, π_1$ of $Symp(M,ω)$ is guided by embedded $J$-holomorphic curves. Further, we prove that for any fixed sizes blowups, when the area ratio $μ$ between the section and fiber goes to infinity, there is a topological colimit of $Symp(M,ω_μ).$ Moreover, when the blowup sizes are all equal to half the area of the fiber class, we give a topological model of the colimit which induces non-trivial symplectic mapping classes in $Symp(M,ω) \cap \rm Diff_0(M),$ where $…
In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along the way, we need to compute the homotopy groups of the compactly supported symplectomorphism groups of the cotangent bundle of $\mathbf{RP}^{2}$ and of $\mathbf{C}^*\times\mathbf{C}$. We also make progress in the case of the $A_n$-Milnor fibres: here we can show that the (compactly supported) Hamiltonian group is contractible and that the symplectic mapping class group embeds in the braid group on $…
Two commuting symplectomorphisms of a symplectic manifold give rise to actions on Floer cohomologies of each other. We prove the elliptic relation saying that the supertraces of these two actions are equal. In the case when a symplectomorphism $f$ commutes with a symplectic involution, the elliptic relation provides a lower bound on the dimension of $HF^*(f)$ in terms of the Lefschetz number of $f$ restricted to the fixed locus of the involution. We apply this bound to prove that Dehn twists around vanishing Lagrangian spheres inside most hypersurfaces in Grassmannians have infinite order in t…
There are two main results. The first states that isotropy subgroups of groups acting transitively on a rationally hyperbolic spaces have infinitely generated rational cohomology algebra. Using this fact, we prove that the analogous statement holds for groups of symplectomorphisms of certain blow-ups.
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.
In this paper, we describe explicit relations in the symplectomorphism groups of toric hypersurfaces. To define the elements involved, we construct a proper stack of toric hypersurfaces with compactifying boundary representing toric hypersurface degenerations. Our relations arise through the study of the one dimensional strata of this stack. The results are then examined from the perspective of homological mirror symmetry where we view sequences of relations as maximal degenerations of Landau-Ginzburg models. We then study the B-model mirror to these degenerations, which gives a new mirror sym…
In this paper we apply Donaldson's general moment map framework for the action of a symplectomorphism group on the corresponding space of compatible (almost) complex structures to the case of rational ruled surfaces. This gives a new approach to understanding the topology of their symplectomorphism groups, based on a result of independent interest: the space of compatible integrable complex structures on any symplectic rational ruled surface is (weakly) contractible. We also explain how in general, under this condition, there is a direct relationship between the topology of a symplectomorphism…
For $(\mathbb{C} P^2 \# 5{\overline {\mathbb{C} P^2}},ω)$, let $N_ω$ be the number of $(-2)$-symplectic spherical homology classes.We completely determine the Torelli symplectic mapping class group (Torelli SMCG): the Torelli SMCG is trivial if $N_ω>8$; it is $π_0(Diff^+(S^2,5))$ if $N_ω=0$ (by Paul Seidel and Jonathan Evans); it is $π_0(Diff^+(S^2,4))$ in the remaining case. Further, we completely determine the rank of $π_1(Symp(\mathbb{C} P^2 \# 5{\overline {\mathbb{C} P^2}}, ω)$ for any given symplectic form. Our results can be uniformly presented regarding Dynkin diagrams of type $\mathbb{…
Let $(X,ω)$ be a symplectic rational 4 manifold. We study the space of tamed almost complex structures $\mathcal{J}_ω$ using a fine decomposition via smooth rational curves and a relative version of the infinite-dimensional Alexander duality. This decomposition provides new understandings of both the variation and stability of the symplectomorphism group $Symp(X,ω)$ when deforming $ω$. In particular, we compute the rank of $π_1(Symp(X,ω))$ with Euler number $χ(X)\leq 7$ in terms of the number $N$ of -2 symplectic sphere classes.
In this paper, we prove homological stability of symplectomorphisms and extended hamiltonians of surfaces made discrete. We construct an isomorphism from the stable homology group of symplectomorphisms and extended Hamiltonians of surfaces to the homology of certain infinite loop spaces. We use these infinite loop spaces to study characteristic classes of surface bundles whose holonomy groups are area preserving, in particular we give a homotopy theoretic proof of the Kotschick-Morita theorem.
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.
Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We show that for certain closed symplectic manifolds, including complex projective spaces and Grassmannians, the universal cover of the group of Hamiltonian symplectomorphisms of the manifold has infinite commutator length. In particular, we present explicit examples of elements in …
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.…
We show that simply connected contact manifolds that are subcritically Stein fillable have a unique symplectically aspherical filling up to diffeomorphism. Various extensions to manifolds with non-trivial fundamental group are discussed. The proof rests on homological restrictions on symplectic fillings derived from a degree-theoretic analysis of the evaluation map on a suitable moduli space of holomorphic spheres. Applications of this homological result include a proof that compositions of right-handed Dehn twists on Liouville domains are of infinite order in the symplectomorphism group. We a…
In this paper we prove that the Torelli part of the symplectomorphism groups of the $n$-point ($n\leq 4$) blow-ups of the projective plane is trivial. Consequently, we determine the symplectic mapping class group. It is generated by reflections on $K_ω- $spherical class with zero $ω$ area.
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…
The Gromov-Eliashberg theorem says that the group of symplectomorphisms of a symplectic manifold is C^0-closed in the group of diffeomorphisms. This can be translated into a statement about the Lagrangian submanifolds which are graphs of symplectomorphisms. It is also known that such Lagrangian submanifolds are locally microsupports of sheaves. We explain how we can deduce the Gromov-Eliashberg theorem from the involutivity theorem of Kashiwara and Schapira which says that the microsupport of a sheaf is coisotropic.
By a theorem of Banyaga the group of diffeomorphisms of a manifold $P$ preserving a regular contact form $α$ is a central $S^1$ extension of the commutator of the group of symplectomorphisms of the base $B = P/S^1$. We show that if $T$ is a Hamiltonian maximal torus in the group of symplectomorphism of $B$, then its preimage under the extension map is a maximal torus not only in the group $\Diff(P, α)$ of diffeomorphisms of $P$ preserving $α$ but also in the much bigger group of contactomorphisms $\Diff (P, ξ)$, the group of diffeomorphism of $P$ preserving the contact distribution $ξ= \ker α$…
In principle, Floer theory can be extended to define homotopy invariants of families of equivalent objects (e.g. Hamiltonian isotopic symplectomorphisms, 3-manifolds, Legendrian knots, etc.) parametrized by a smooth manifold B. The invariant of a family consists of a filtered chain homotopy type, which gives rise to a spectral sequence whose E^2 term is the homology of B with twisted coefficients in the Floer homology of the fibers. The filtered chain homotopy type also gives rise to a "family Floer homology" to which the spectral sequence converges. For any particular version of Floer theory,…
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…
Making use of the extended flux homomorphism on the group of symplectomorphisms of a closed oriented surface of genus at least 2, we introduce new characteristic classes of foliated surface bundles with symplectic, equivalently area-preserving, total holonomy. These characteristic classes are stable with respect to the genus and we show that they are highly non-trivial. We also prove that the second homology of the group of Hamiltonian symplectomorphisms equipped with the discrete topology is very large.
Consider a group $Γ$ acting on a formal (Fedosov) deformation quantization $\mathbb{A}_\hbar(M)$ of a symplectic manifold $(M,ω)$. This canonically induces an action of $Γ$ by symplectomorphisms on $M$. We examine the reverse problem of extending group actions by symplectomorphisms to the deformation quantization. To do this we first define a notion of extension that does not impose restrictions on the Fedosov connection realizing $\mathbb{A}_\hbar(M)$ in its gauge equivalence class by considering composition of pull-back with certain inner automorphisms of sections of the Weyl bundle. We refo…
In an earlier paper we explained how to convert the problem of symplectically embedding one 4-dimensional ellipsoid into another into the problem of embedding a certain set of disjoint balls into \CP^2 by using a new way to desingularize orbifold blow ups Z of the weighted projective space \CP^2_{1,m,n}. We now use a related method to construct symplectomorphisms of these spaces Z. This allows us to construct some well known Fano 3-folds (including the Mukai--Umemura 3-fold) in purely symplectic terms, using a classification by Tolman of a particular class of Hamiltonian S^1-manifolds. We also…
The Delzant theorem of symplectic topology is used to derive the completely integrable compactified Ruijsenaars-Schneider III(b) system from a quasi-Hamiltonian reduction of the internally fused double SU(n) x SU(n). In particular, the reduced spectral functions depending respectively on the first and second SU(n) factor of the double engender two toric moment maps on the III(b) phase space CP(n-1) that play the roles of action-variables and particle-positions. A suitable central extension of the SL(2,Z) mapping class group of the torus with one boundary component is shown to act on the quasi-…
