SYMPLECTIC CONFIGURATIONS

ŚWIAT GAL AND JAREK KȨDRA

Abstract. We define a class of symplectic fibrations called symplectic configurations. They are natural generalization of Hamiltonian fibrations. Their geometric and topological properties are investigated. We are mainly concentrated on integral symplectic manifolds.

We construct the classifyng space � of symplectic integral configurations. The properties of the classifying map � → BSymp(M, ω) are examined. The universal symplectic bundle over � has a natural connection whose holonomy group is the enlarged Hamiltonian group recently defined by McDuff.

The space � is identified with the classifying space of an extension of certain subgroup of the symplectomorphism group.

Contents

1. Introduction ........................................................1 2. Topology of symplectic configurations .............................. 6 3. Geometry of symplectic configurations ............................ 11 4. Group cohomology and the group of McDuff ...................... 16 5. Cohomology ring of symplectic configurations.....................21

1. Introduction

1.1. Symplectic configurations. The subject of this paper is certain class of symplectic fibrations called symplectic configurations. They are defined as follows.

Definition 1.1. Let (M, ω) and (W, ω_{W}) be symplectic manifolds. We say that a symplectic fiber bundle (fiber bundle with a structure group Symp(M, ω)) M → E → B is a W-symplectic configuration if there exists a map E → W such that when restricted to any fiber of E is a symplectic embedding. If W is integral we say that E is an integral configuration. The fibration E is called an (integral) symplectic configuration if it is a W-configuration for some (integral) (W, ω_{W}).

Here is an alternative approach. Consider the space Symp(M, W) of symplectic embeddings f: (M, ω) → (W, ω_{W}). The group of all symplectomorphisms of the source acts freely on that space and the quotient is denoted by �_{W}. We call it the space of symplectic configurations of (M, ω) in (W, ω_{W}). We get a principal fibration

Symp(M, ω) → Symp(M, W) → �_{W}

Date: April 21, 2005. 2000 Mathematics Subject Classification. Primary 53D, 55R; Secondary 53C. Key words and phrases. symplectic fibration; h-principle; classifying space; The first author is partially supported by the State Committee for Scientific Research (grant no. 2

P03A 017 25) and by the Foundation for Polish Science.

The second author is partially supported by the State Committee for Scientific Research (grant no. 1 PO3A 023 27).

and the associated symplectic one

(M, ω) → M_{W} → �_{W}.

Proposition 1.2. Symplectic fibration E is a W-symplectic configuration if and only if it is a pull-back of the bundle M_{W}.

Proof. Let E → W be a map such that it is a symplectic embedding on every fiber. Clearly, it defines a map B → �_{W} such that E is a pull-back of M_{W}.

To prove the converse choose a point p ∈ M and consider the following composition of two maps:

Symp(M, W) −−−⟶^{𝒫} M_{W} −−−⟶^{ev} W,

where

𝒫(f): = [f, p] and ev[f, q] = f (q).

The map ev is well defined. Indeed M_{W} is the quotient of Symp(M, W) × M by the relation [f ∘ ψ, q] ∼ [f, ψ(q)] and ev([f ∘ ψ, q]) = f (ψ(q)) = ev([f, ψ(q)]). The evaluation map is the required map for the fibration M_{W}. Since E is a pull-back of M_{W} then the proof is finished.

1.2. Closed connection 2-form. Without loss of generality we will assume that the base of a fibration is a manifold thus we may speak of the differential forms on E.

Recall that if (M, ω) → E → B is a symplectic fibration then a closed two form Ω ∈ Ω^{2}(E) is called a closed symplectic connection 2-form if it restricts to the symplectic form on any fiber (see Lemma 6.18 in [12] for the general definition). According to Thurston (Theorem 6.3 in [12]), the existence of a closed extension is purely cohomological condition, i.e. the form Ω exists if and only if the class [ω] belongs to the image of the map H^{2}(E; ) → H^{2}(M; ).

The existence of a closed connection 2-form is important. It was first examined by Guillemin, Lerman and Sternberg [5] where they introduced the notion of a coupling form. It is a certain closed connection 2-form satisfying some uniqueness condition. Further, the fiber integrals of powers of the cohomology class of the coupling form gives a sequence of symplectic characteristic classes [7, 8].

Lemma 1.3. A symplectic bundle (M, ω) → E → B is a symplectic configuration if and only if it admits a closed connection form.

Proof. If f: E → W defines a configuration, then f^{∗}(ω_{W}) is a closed connection 2-form on E.

To proof of the converse construct a map f : (E, Ω) → (W, ω_{W}) := _{i}( ^{n_{i}}, a_{i}ω_{n_{i}}) such that f^{∗}[ω′] = [Ω] for appropiate a_{i} ∈ , where ω_{n} is the standard symplectic form on ^{n}. To see that the map f exists express [Ω] as a linear combination ∑ _{i} a_{i}η_{i} of integral classes. The classifying map for a tuple (η_{1}, ..., η_{m}) has _{i}( ^{∞} , a_{i}ω_{∞} ) as a target and factrors through a product of finite dimensional projective spaces as required. For sufficiently lagre ∑ n_{i} we may perturb f into an embedding preserving the 2-form, according to the h-principle (see Section 3.4.2 in [4]).

1.3. Universal integral configuration. A symplectic manifold (M, ω) is called integral if [ω] is in the image of the map H^{2}(M; ) → H^{2}(M, ). Due to Gromov, every integral symplectic manifold embeds into complex projective space ( ^{N}, ω_{N}) equipped with the standard symplectic form for N big enough (Exercise 3.4.2.(1) in [4]). Spaces of symplectic embeddings satisfy some obvious functoriality properties and we define Symp(M, ^{∞} ): = ⋃ _{N→∞} Symp(M, ^{N}). Its quotient by Symp(M, ω) is denoted by �_{∞} and called the universal symplectic configuration space. As before we have the universal principal fibration

Symp(M, ω) → Symp(M, ^{∞} ) → �_{∞}

and the associated one

(M, ω) → M_{�_{∞}} → �_{∞} .

Let (M, ω) → E → B be a symplectic fibration whose classifying map ξ: B → BSymp(M, ω) has a lift ξ̂: B → �_{∞} . This lift defines a map E → M_{�_{∞}} →^{ev} ^{∞} which factors through ^{N}, since E is finite dimensional. Thus E is an integral configuration (see Definition 1.1), and the pull-back of the standard symplectic form ω_{N} on ^{N} is a closed integral connection form on E.

1.4. Torsion. The space Symp(M, ^{∞} ) has the same number of connected components as the set of maps from M to ^{∞} such that the generator of H^{2}( ^{∞} , ) pulls back to [ω]. The homotopy classes of maps from M to ^{∞} are in one-to-one correspondence with H^{2}(M, ). The kernel of the map H^{2}(M, ) → H^{2}(M, ) consists of the torsion of H^{2}(M, ). Thus if this torsion in nontrivial then the space Symp(M, ^{∞} ) is disconnected. If M is compact then Symp(M, ^{∞} ) has finite number of connected components. Moreover, if Symp(M, ω) acts nontransitively on the preimage of ω in H^{2}(M, ) then � is disconnected.

Let us observe that any (finite) Abelian group T can be the torsion of H^{2}(M; ). First notice that the torsion of H^{2}(M; ) is isomorphic to the torsion of H_{1}(M; ) (see Corollary 5.5.4 in [15]). Take finitely presented group G such that its abelianization has torsion equal to T (e.g. G = T). Due to Gompf [3], there exists a closed symplectic 4-manifold (M, ω) with fundamental group G. Hence the torsion of H^{2}(M; ) is isomorphic to T.

Let (M, ω) be an integral symplectic manifold and let ω̂ denote a pair (ω, τ) of the symplectic form and a lift of its class to H^{2}(M, ). Define

Symp(M, ω̂): = {ψ ∈ Symp(M, ω): ψ^{∗}(τ) = τ}

to be an open-closed (finite index if M is compact) subgroup of Symp(M, ω) consisting of the symplectomorphisms which preserve the class τ. Notice that if H^{2}(M, ) is torsion-free then Symp(M, ω̂) = Symp(M, ω).

Lemma 1.4. If (M, ω) → E → B is an integral symplectic configuration, then there exists a lift τ ∈ H^{2}(M, ) of [ω] such that the structure group of E reduces to Symp(M, ω̂).

Proof. If f: E → ^{∞} defines a configuration structure on E then the pull-back of the generator of H^{2}( ^{∞} , ) has the above property.

Let Symp_{τ}(M, ^{∞} ) denote the connected component of Symp(M, ^{∞} ) containing symplectic embeddings corresponding to τ ∈ H^{2}(M, ). The quotient � = �_{τ} = Symp_{τ}(M, ^{∞} )/ Symp(M, ω̂) is the component of �_{∞} corresponding to τ. In the sequel, we restrict ourselves to the investigation of �. We will also call it a universal configuration space. Again we have universal fibrations:

Symp(M, ω̂) → Symp_{τ}(M, ^{∞} ) → �

and the associated one

(M, ω) → M_{�} → �.

A symplectic fibration whose classifying map lifts to �_{τ} we call an integral symplectic configuration compatible with τ. Notice that a symplectic integral configuration might be compatible with several lifts.

Since Symp(M, ω̂) is open-closed in Symp(M, ω) then their connected components of the identity are equal. Therefore the map of classifying spaces induced by the inclusion is a covering. In particular the higher homotopy groups of these classifying spaces are equal as well as the rational (co-) homology.

1.5. Homotopy properties of the universal configuration space. Integral symplectic configurations constitute a big class of symplectic fibrations. To see this consider the classifying map

ℱ: � → BSymp(M, ω)

and the maps ℱ_{k}: π_{k}(�) → π_{k}(BSymp(M, ω)) induced on homotopy groups. The following theorem is proven in Section 2.2.

Theorem 1.5. Let (M, ω) be a compact integral symplectic manifold. Then

(1) π_{k}(�) ^{ℱ_{k}} π_{k}(BSymp(M, ω)) for k 1, 2.

(2) The following sequences are exact.

0 → H^{0}(M; ) → π_{2}(�)  →^{ℱ_{2}} π_{2}(BSymp(M, ω)) → Γ_{ω} → 0
0 → Γ_{ω} → H^{1}(M; ) → π_{1}(�)  →^{ℱ_{1}} π_{1}(BSymp(M, ω̂)) → 0

1.6. An extension of Symp(M, ω̂). We identify the universal symplectic configuration space � as the classifying space B𝒢_{τ} of certain extension 𝒢_{τ} of Symp(M, ω̂). More precisely this extension is of the form

0 → Map(M, U(1)) → 𝒢_{τ} → Symp(M, ω̂) → 1,

where Map(M, U(1)) is the (Abelian) gauge group of continuous maps from M to

U(1) (see Section 2.3 for the precise definition). We shall usually omit the subscript τ if it does not lead to a confusion. The proof of the next theorem is in Section 2.3.

Theorem 1.6. The classifying space of 𝒢 is homotopy equivalent to the universal symplectic configuration space:

B𝒢 = �.

Moreover, the classifying map ℱ: � = B𝒢 → BSymp(M, ω̂) is induced by the projection 𝒢 → Symp(M, ω̂).

1.7. Holonomy group. If Ω is the symplectic form on ^{∞} and ev: M_{�} → ^{∞} is the evaluation map, then ev^{∗}Ω is the closed connection form on the universal symplectic configuration (M, ω) → M_{�} → �. It defines a symplectic connection whose holonomy group is denoted by 𝒟 = 𝒟_{τ} ⊂ Symp(M, ω̂).

Note that 𝒟 depends on the lift τ and is defined only up to conjugacy. In general we will omit the subscript τ except the necessary cases.

Theorem 1.7. (1) 𝒟 intersects every connected component of Symp(M, ω̂); (2) The identity component 𝒟_{0} of 𝒟 is equal to Ham(M, ω);

(3) 𝒟 ∩ Symp_{0}(M, ω) = Flux^{−1}(H^{1}(M; )/Γ_{ω}) that is the intersection consists of symplectomorphisms with integral flux.

Proof. (1) Since π_{1}(�) → π_{1}(B Symp(M, ω̂)) is a surjection (Theorem 1.5) then 𝒟 intersects every connected component of Symp(M, ω̂).

(2) Let (M, ω) → M_{W} → �_{W} be a symplectic configuration fibration. Since Ω_{W} is a closed connection 2-form then the holonomy about any contractible loop is Hamiltonian due to Theorem 6.21 in [12]. This proves that 𝒟_{0} ⊂ Ham(M, ω). The converse inclusion follows from Corollary 3.4.

(3) This part will be proved in Proposition 3.7(2).

Finally we find a connection between the holonomy subgroup 𝒟 and the extension 𝒢.

Theorem 1.8. Let Map_{0}(M, U(1)) denote the identity component of Map(M, U(1)). There exists a commutative triangle of continuous homomorphisms

\xymatrix{ & 𝒢/ Map_{0}(M, U(1)) \ar[d] \\ 𝒟 \ar[r] \ar[ur] & Symp(M, ω̂)}

in which the diagonal arrow is a homotopy equivalence. In particular, the structure group of E lifts to 𝒢/ Map_{0}(M, U(1)) if and only if the structure group of E reduces to 𝒟.

The next two theorems summarize all main results of the paper.

Theorem 1.9. Let (M, ω) be a closed integral symplectic manifold and let τ be an integral lift of [ω]. Let (M, ω) → E → B be a symplectic fibration. The following conditions are equivalent:

(1) E admits a closed integral connection 2-form Ω compatible with τ, i.e. there exists 𝒯 ∈ H^{2}(E, ) such that it lifts Ω and extends τ;

(2) E is an integral symplectic configuration compatible with τ;

(3) the structure group of E lifts to 𝒢_{τ}.

Proof. (1) ⇒ (2) is proved in Section 2.4,

(1) ⇒ (3) is proved in Section 2.5,

(2) ⇒ (1) follows from Lemma 1.3,

(2) ⇔ (3) follows form Theorem 1.6,

Clearly, if E is an integral symplectic configuration with respect to the lift τ then its structure group reduces to the holonomy group 𝒟_{τ}. In other words, if the structure group lifts to 𝒢_{τ} then it lifts to 𝒢_{τ}/ Map_{0}(M, U(1)). The folowing example due to McDuff shows that the converse in not true.

Example 1.10. Let S^{2} → BSO(2) →^{π} BSO(3) be the universal fibration associated with the action of SO(3) on S^{2}. Let τ be the generator of H^{2}(S^{2}, ). We claim that τ does not admit an integral extension although the fibration is Hamiltonian. Indeed, H^{3}(BSO(2); ) = 0 therefore H^{3}(BSO(3), ) = /2 is in the kernel of π^{∗}. Hence the generator of the latter group is killed by the generator τ ∈ H^{2}(S^{2}, ) in the spectral sequence. Thus τ is not in the image of π^{∗} : H^{2}(BSO(2)) → H^{2}(S^{2}, ) i.e. it does not admit an integral extension.

The proof of the following theorem will be given in Section 3.4.

Theorem 1.11. Suppose that the structure group of a symplectic fibration M → E → B reduces to 𝒟_{τ} ⊂ Symp(M, ω̂). If H_{2}(B, ) is torsion-free then E is an integral configuration.

Remark 1.12. We do not prove that E is a configuration compatible with τ. We show that d_{2}(τ) = 0 and d_{3}(τ) ∈ E_{3}^{3,0} is torsion in the spectral sequence assocoated with M → E → B. Thus a priori τ might not extend. Notice that, although E_{2}^{3,0} = H^{3}(B, ) is (by the assumption) torsion free, E_{3}^{3,0} = E_{2}^{3,0}/d_{2}(E_{2}^{1,1}) might have nontrivial torision. However, we don’t know any example in which E is a configuration not compatible with τ.

1.8. Other results.

(1) Section 3 is devoted to the geometry of symplectic configurations. More precisely, we define a principal connection on fibrations Symp(M, W) → �_{W} and investigate its properties.

(2) In Section 4, we investigate the group cohomology relations between groups 𝒟 and 𝒢. We also compare these groups with certain subgroups Ham^{s} (M, ω) of Symp(M, ω) recently discovered by McDuff [11].

(3) In Section 5, we investigate characteristic classes of integral symplectic configurations and relate them to other known characteristic classes.

1.9. Acknowledgement. The idea of configurations grew out of discussions with Tadeusz Januszkiewicz.

We warmly thank Dusa McDuff for showing us the early version of the paper [11], discussions, comments and picking up mistakes and inaccuracies of this paper. In particular, she drawed our attention to the torsion issues.

We thank Agnieszka Bojanowska, Stefan Jackowski and Michael Weiss for homotopy discussions and hospitality.

2. Topology of symplectic configurations

2.1. Gromov’s h-principle. Let Map(M, ^{∞} ) (respectively Map^{∞} (M, ^{∞} ) ) denotes the space of all continuous (respectively smooth) maps from M to ^{∞} equipped with compact-open (respectively C^{∞} ) topology. The space of smooth maps is defined as ⋃ _{n} C^{∞} (M, ^{n}). It is well known that the inclusion Map^{∞} (M, ^{∞} ) → Map(M, ^{∞} ). induces a (weak) homotopy equivalence.

Theorem 2.1. Let τ ∈ H^{2}(M, ) be a lift of [ω] ∈ H^{2}(M, ). Let Symp_{τ}(M, ^{∞} ) and Map_{τ}(M, ^{∞} ) denote the connected components corresponding to the class τ. The inclusion i: Symp_{τ}(M, ^{∞} ) → Map_{τ}(M, ^{∞} ) induces an isomorphisms on homotopy groups.

Remark 2.2. The above theorem easily follows from the parametric version of hprinciple for symplectic embeddings. Unfortunately, it seems that there is no proof of it in the literature yet. That is why we prove the above theorem appealing only to the well known Gromov’s h-principle for symplectic immersions and its parametric version (Theorem (A) and Exercise (2) in Section 3.4.2 of [4]).

Proof of Theorem 2.1.Since the spaces of smooth and continuous maps are weakly homotopy equivalent we can assume that all maps are smooth.

Injectivity: Suppose that s ∈ ker i_{∗} that is there exist a commutative diagram

\xymatrix{S^{k} × M \ar[r]^{f} \ar[d] & ^{m} \\ D^{k+1} × M \ar[ur]^{F}}

where f is symplectic embedding on each fiber s × M and F is smooth embedding on fibers and equals f over the boundary. The argument consists of several steps.

We first deform F to a fiberwise symplectic immersion F′ so that it equals f over the boundary. This can be done according to the parametric h-principle for symplectic immersions (Theorem 16.4.3 in [2]). We want to improve it so that it will be fiberwise symplectic embedding.

Let j : M → ^{N} ⊂ ^{N} be the composition of an embedding if M into ^{N} and the standard inclusion of Lagrangian ^{N} into ^{N}. Then j is an isotropic embedding. Moreover, let φ : ^{m} × D^{2N} → ^{l} be a symplectic embedding that is linear on m × 0. Take

D^{k+1} × M → ^{m} × D^{2N} → ^{l}

defined by

(d, m) ↦ φ(F′(d, m), α(d)j(m)),

where α : D^{k+1} → is a scaling function such that it equals 0 exactly on the boundary sphere. Clearly, it is an embedding because j is. It is symplectic because F′ is fiberwise symplectic and j is isotropic. Surjectivity: We shall show that every fiberwise smooth embedding f : S^{k} × M → ^{m} (fiberwise in Map_{τ}(M, ^{m})) is homotopic to a fiberwise symplectic embedding. First take the composition F : T^{∗}S^{k}× M → ^{m} of f and the projection T^{∗}S^{k} × M → S^{k} × M. According to the standard h-principle there exist arbitrarily small deformation of F that is a symplectic embedding (with respect to the product symplectic form). The restriction of it to the image of the “zero section” S^{k} × M ⊂ T^{∗}S^{k}× M gives a symplectic embedding. The above homotopy might not preserve the basepoint but this is not a problem due to the injectivity of i_{∗}.

2.2. Proof of Theorem 1.5.

Lemma 2.3. π_{k}Map(M, ^{∞} ) = H^{2−k}(M, ), where we set H^{m}(M, ) = 0 for m < 0.

Proof. If follows from the basic algebraic topology (e.g. [15]). Let s ∈ S^{k} and f ∈ Map(M, X) be a base point, the constant map m ↦ x ∈ X. Moreover let M_{+} denote M with artificially added base point. Then we calculate

π_{k}(Map(M, X)) = [(S^{k}, s), (Map(M, X), f)]
= [M_{+} ∧ S^{k}, (X, x)]
= [M_{+}, Ω^{k}(X, x)]

Since ^{∞} = K( , 2) we have that π_{k}(Map(M, ^{∞} )) = [M, K( , 2−k)] = H^{2−k}(M, ).

It is easy to see a composition of this isomorphism with the projection H^{2−k}(M, ) → Hom(H_{2−k}(M, ), ) (which is an isomorphism for k ∈ {1, 2}) as follows. Let [ξ] ∈ π_{k}(Map(M, ^{∞} )) be represented by a map ξ: S^{k} × M → ^{∞} . The value of the resulting cohomology class on (2 − k)−cycle a is equal to ⟨[S^{k} × a], ξ^{∗}ω_{∞} ⟩, where ω_{∞} ∈ H^{2}( ^{∞} ; ) is the generator.

Corollary 2.4. π_{k}Symp_{τ}(M, ^{∞} ) = \{ \begin{matrix}H^{2−k}(M; ) & for k ∈ \{1, 2\} \\ 0 & otherwise \end{matrix}

and hence π_{k}(�) π_{k}(BSymp(M, ω̂)) for k > 3.

Proof. Notice that Map(M, ^{∞} ) is an H-space (since ^{∞} is). Therefore all its connected components are homotopy equivalent. The first statement follows immediately from Theorem 2.1 and Lemma 2.3. The second is the direct application of the long exact sequence of homotopy group for the fibration Symp(M, ω̂) → Symp_{τ}(M, ^{∞} ) → �.

Example 2.5. Let Σ_{g} be a surface of genus g > 0. Let i_{0}, i_{1} ∈ Symp(Σ_{g}, ^{∞} ) be two embeddings such that i_{0} = i_{1} ∘ f, where f: Σ_{g} → Σ_{g} is a Dehn twist. One can construct a symplectic Lefschetz fibration (M, ω) → S^{2} with the generic fiber Σ_{g} such that the monodromy about some critical value is exactly f and (M, ω) is an integral symplectic manifold [1]. Embedding (M, ω) into ^{n} we also get a family of symplectic embeddings i_{t}: Σ_{g} → ^{N} for t ∈ [0, 1]. Since π_{0}(Symp(Σ_{g}, area)) is generated by Dehn twists, then we obtain that the map π_{0}(Symp(Σ_{g}, area)) → π_{0}(Symp(Σ_{g}, ^{∞} )) is trivial.

Lemma 2.6. The map π_{2}(Symp(M, ω)) → π_{2}(Symp(M, ^{∞} )) is trivial. In particular, π_{3}(�) π_{3}(BSymp(M, ω)).

Proof. Let ξ ∈ π_{2}(Symp(M, ω)). Its image in π_{2}(Map(M, ^{∞} )) = is equal to ⟨[ω], ev_{∗}(ξ)⟩, where ev: Symp(M, ω) → M is the evaluation at the base point pt ∈ M (cf. proof of Lemma 2.3). More precisely, consider the following commutative diagram.

\xymatrix{Symp(M, ω) \ar[r]^{i} \ar[d]^{ev} & Symp(M, ^{∞} ) \ar[d]^{ev} \\ M \ar[r]^{f} & ^{∞}}

The isomorphism π_{2}(Map(M, ^{∞} )) = is given by the integration of the generator [ω_{∞} ] ∈ H^{2}( ^{∞} ) over the sphere S^{2} → Map(M, ^{∞} ) →^{ev} ^{∞} . Hence we calculate that ⟨[ω_{∞} ], ev_{∗}i_{∗}(ξ)⟩= ⟨[ω_{∞} ], f_{∗}ev_{∗}(ξ)⟩ = ⟨f^{∗}[ω_{∞} ], ev_{∗}(ξ)⟩ = ⟨[ω], ev_{∗}(ξ)⟩as required.

We claim that this number is equal to zero. Indeed, if [ω] is a sum of products of 1dimensional classes then ⟨[ω], ev_{∗}(ξ)⟩= ⟨ξ^{∗}(ev^{∗}[ω]), [S^{2}]⟩ = 0. Suppose that [ω] is not a sum of products. Recall that, according to Milnor and Moore [13], the cohomology ring of Symp(M, ω) (which is an H-space) is free graded commutative algebra. If ev^{∗}[ω] were nonzero then it would be of infinite order, which is impossible since [ω]^{n+1} = 0.

Let Flux : π_{1}(Symp(M, ω)) → H^{1}(M; ) be the flux homomorphism defined as follows. ⟨Fluxξ, γ⟩ = ⟨[ω], ξ_{∗}(γ)⟩, where ξ_{∗}(γ): ^{2} → M sends (t, s) → ξ_{t}(γ(s)) (see Chapter 9 in [12] for details). Its image is called the flux group and is denoted by Γ_{ω}. Notice that in the case of integral symplectic structure Γ_{ω} is a subgroup of the integral cohomology H^{1}(M, ). The following lemma is an immediate consequence of the above definition.

Lemma 2.7. The following diagram is commutative.

π_{1}(Symp(M, ω)) −−−⟶  π_{1}(Symp(M, ^{∞} ))
Flux↓  ^{}↓
Γ_{ω}  −−−⟶ H^{1}(M, ) = π_{1}(Map(M, ^{∞} ))

Proof of Theorem 1.5.

(1) This part follows from Corollary 2.4 and Lemma 2.6.

(2) To prove that the first sequence is exact one applies Lemma 2.6 and 2.7 to the exact sequence associated with the fibration Symp(M, ω̂) → Symp_{τ}(M, ^{∞} ) → �. Similarly, Corollary 2.4 and Lemma 2.7 applied to the long exact sequence of the the same fibration proves the exactness of the second sequence.

2.3. Proof of Theorem 1.6. Recall that G-principle bundles over M are classified by H^{1}(M, G) and H^{1}(M, U(1)) is canonically isomorphic to H^{2}(M, ). The bundle corresponding to τ ∈ H^{2}(M, ) will be denoted L_{τ}, and the class of a bundle L in H^{2}(M, ) we will be denoted by [L].

Assume that (M, ω) and (W, ω_{W}) are integral symplectic manifolds. Choose τ ∈ H^{2}(M, ) and τ_{W} ∈ H^{2}(W, ) to be some lifts of ω and ω_{W} respectively. Let

Map_{Symp}^{U(1)}(L_{τ}, L_{τ_{W}})

be the space of maps from L_{τ} to L_{τ_{W}} which are U(1)-equivariant and cover a symplectomorphism (M, ω) → (W, ω_{W}):

L_{τ} −−−⟶^{ψ̂} L_{τ_{W}}
↓  ↓
M −−−⟶^{ψ} W.

Observe that if ψ: M → W is a continuous map, then ψ^{∗}(L_{τ}) = L_{ψ^{∗}τ}, thus there exists a covering U(1)-equivariant map f̂: L_{ψ^{∗}τ} → L_{τ}.

In particular, for (W, ω_{W}) = (M, ω) we get a group

𝒢: = Map_{Symp}^{U(1)}(L_{τ}, L_{τ}).

Lemma 2.8. There is an extension

1 → Map(M, U(1)) → 𝒢 → Symp(M, ω̂) → 1

Proof. The kernel of the map that assigns to ψ̃the underlying symplectomorphism ψ consists of the automorphisms which cover the identity. Over each point of M such automorphism is given by an element of U(1).

Proposition 2.9. Let ω_{n} denote the standard symplectic form on ^{n}, and let [L_{n}] = [ω_{n}] ∈ H^{2}( ^{n}, ). Then the space

Map_{Symp}^{U(1)}(L_{τ}, L_{∞} ): = ⋃_{n} Map_{Symp}^{U(1)}(L_{τ}, L_{n})

is contractible.

Proof. Let Map^{U(1)}(L_{τ}, L_{∞} ) be a space of all U(1)-equivariant maps L_{τ} → L_{∞} . Then we have a bundle:

Map(M, U(1)) → Map^{U(1)}(L_{τ}, L_{∞} ) → Map_{τ}(M, ^{∞} ),

Consider the following two exact sequences of homotopy groups

\xymatrix{…π_{k}(Map(M, U(1))) \ar[r] & π_{k}(Map^{U(1)}(L_{τ}, L_{∞} )) \ar[r] & π_{k}(Map_{τ}(M, ^{∞} )) … \\ …π_{k}(Map(M, U(1))) \ar[r] \ar[u]^{=} & π_{k}(Map_{Symp}^{U(1)}(L_{τ}, L_{∞} )) \ar[r] \ar[u] & π_{k}(Symp_{τ}(M, ^{∞} )) … \ar[u]}

where the vertical arrows are induced by inclusions. According to Theorem 2.1, the map

π_{k}(Symp_{τ}(M, ^{∞} )) → π_{k}(Map_{τ}(M, ^{∞} ))

is an isomorphism. By the five lemma the same is true for

π_{k}(Map_{Symp}^{U(1)}(L_{τ}, L_{∞} )) → π_{k}(Map^{U(1)}(L_{τ}, L_{∞} )).

We are left to show that Map^{U(1)}(L_{τ}, L_{∞} ) is contractible. Since L_{∞} = EU(1), then the following lemma finishes the proof.

Lemma 2.10. Let E be a free G space. Then Map^{G}(E, EG) is contractible.

Proof. It is a standard fact (about universal principal bundles) that Map^{G}(E, EG) is connected (see Theorem 8.12 on page 58 in [16]). We need to show that for all k the space Map(S^{k}, Map^{G}(E, EG)) = Map^{G}(E × S^{k}, EG) is connected. But S^{k} × E is again a free G space and we apply the observation we began with.

Proof of Theorem 1.6.Consider the following two fibrations

Map(M, U(1)) → Map_{Symp}^{U(1)}(L_{τ}, L_{∞} ) →^{p_{1}} Symp_{τ}(M, ^{∞} )

and

Symp(M, ω̂) → Symp_{τ}(M, ^{∞} ) →^{p_{2}} �_{τ},

where the base �_{τ} is defined to be this quotient. Observe that the composition p_{2} ∘ p_{1} defines a fibration

𝒢 → Map_{Symp}^{U(1)}(L_{τ}, L_{∞} ) −−−⟶^{p_{2}∘p_{1}} �_{τ}

with a contractible total space.

2.4. Proof of Theorem 1.9 (1 ⇒ 2). By assumption the connection form Ω has a lift 𝒯 ∈ H^{2}(E, ) = [E, K( , 2)]. Since K( , 2) = ^{∞} then we get an embedding f : E → ^{N} such that 𝒯 is the pull-back of the generator. According to to the original Gromov’s h-principle (Theorem 3.4.2 A in [4]) we can deform f to an immersion preserving forms. That is an immersion f′ : E → ^{N} so that f^{∗}(ω_{N}) = Ω. At the price of increasing N we can deform it further to get a embedding as we did in the proof of Theorem 2.1.

2.5. Proof of Theorem 1.9 (1 ⇒ 3). A class 𝒯 defines a line bundle L_{𝒯} → E. Notice that the pull-back bundle under the inclusion of the fiber i: M → E is the line bundle L_{i^{∗}𝒯} → M. Thus i^{∗}𝒯 is a lift of [ω] to H^{2}(M, ).

Consider the composition L_{𝒯} → E → B as a bundle over B with fiber U(1). The structure group of this fibration is contained in 𝒢. Thus we get a classifying map f̂: B → B𝒢 such that the following diagram commutes up to homotopy (here f classifies E):

\xymatrix{ & B𝒢 \ar[d] \\ B \ar[ur]^{f̂} \ar[r]^{f} & BSymp(M, ω)}

Recall that according to Theorem 1.6 B𝒢 ≃ �. The classifying map f̂: B → B𝒢 = � is the one constructed in the previous section.

3. Geometry of symplectic configurations

3.1. A principal connection on Symp(M, W). The main reference for connections is Kobayashi-Nomizu [9]. The tangent space to Symp(M, W) at point f is equal to

\{𝒳 ∈ Γ(f^{∗}TW) : 𝒳(p) = \frac{d}{dt}f_{t}(p) f_{t} ∈ Symp(M, W)\}.

We define the horizontal space ℋ_{f} to be the space of the sections f^{∗}ω_{W}-orthogonal to M:

ℋ_{f} := {𝒳 ∈ T_{f} Symp(M, W) : f^{∗}ω_{W}(𝒳, Y) = 0 ∀Y ∈ Γ(TM)}.

Consider the following 1-form θ on Symp(M, W) with values in closed 1-forms on M which is identified with the Lie algebra of Symp(M, ω)

θ_{f} (𝒳): = f^{∗}(ι_{𝒳_{f}} ω_{W})

for 𝒳 ∈ T_{f} Symp(M, W). More precisely

(θ_{f} (𝒳))(p)(Y) = ω_{W}(𝒳_{f} (p), f_{∗}Y(p)),

where Y is a vector field on M and 𝒳_{f} is a vector field on W defined on f (M).

Lemma 3.1. The 1-form θ is a connection form, that is it satisfies

(1) θ_{f ∘ψ} = Ad_{ψ}θ = ψ^{∗}∘ θ,

(2) θ_{f} (\underline{X}) = X, where X ∈ Ω^{1}(M) is a closed 1-form and \underline{X} denotes the fundamental vector field.

Proof. (1) This is an immediate calculation:

(θ_{f ∘ψ} (𝒳))(p)(Y) = ω_{W}(𝒳_{f ∘ψ} (p), (f ∘ ψ)_{∗}Y(p)
= ω_{W}(𝒳_{f} (ψ(p)), f_{∗}(ψ_{∗}Y(p))
= (θ_{f} (𝒳_{f}))(ψ(p))(ψ_{∗}Y(p))
= (ψ^{∗}∘ θ_{f})(𝒳_{f})(p)(Y).

(2) If X is a closed one form on M then we denote by X^{♯}the ω-corresponding vector field and by ψ_{t} its flow. That is ι_{X^{♯}}ω = X. Thus \underline{X}_{f} = X^{♯} under the identification between M and f (M) ∈ W given by f. Hence we get

θ_{f} (X^{♯}) = f^{∗}(ι_{X}^{♯}ω_{W}) = X.

Proposition 3.2. The connection associated to θ on M_{W} and connection defined by the 2-form Ω_{W}: = ev^{∗}(ω_{W}) coincide.

Proof. Both distributions consist of vectors 𝒳 ∈ T_{[f,p]}M_{W} satisfying the condition ev_{∗}(𝒳) ∈ T_{f (p)}W is ω_{W}-orthogonal to f_{∗}(T_{p}M).

3.2. The curvature of θ. Let’s calculate the curvature 2-form of the connection θ. By definition it is

Θ(𝒳, 𝒴): = dθ(𝒳^{h}, 𝒴^{h}) = −θ([𝒳^{h}, 𝒴^{h}]),

where 𝒳^{h}, 𝒴^{h} ∈ ℋ are horizontal parts of tangent vectors. Calculating it further we get

−Θ_{f} (𝒳^{h}, 𝒴^{h}) = θ_{f} [𝒳^{h}, 𝒴^{h}]
= f^{∗}(ι_{[𝒳^{h},𝒴^{h}]} ω_{W})
= f^{∗}({α_{𝒳} , α_{𝒴} })

Here α_{𝒳} , α_{𝒴} are 1-forms associated with extensions of 𝒳^{h}, 𝒴^{h} ∈ Γ(f^{∗}TW) to vector field on some neighborhood of f (M) ⊂ W and {α_{𝒳} , α_{𝒴} } is the Poisson bracket. Since 𝒳^{h} and 𝒴^{h} are horizontal then the 1-forms α_{𝒳} , α_{𝒴} vanish on M. Therefore their local extensions might be chosen to be exact and we get

−Θ_{f} (𝒳^{h}, 𝒴^{h}) = d [f^{∗}{H_{𝒳} , H_{𝒴} } ],

where H_{𝒳} , H_{𝒴} : W → are Hamiltonians, i.e. dH_{𝒳} = α_{𝒳} and dH_{𝒴} = α_{𝒴}.

In other words the curvature of the connection θ at the embedding f: (M, ω) → (W, ω_{W}) is measured by the subspace of functions on M consisting of restrictions of Poisson brackets of functions on W which are constant on f (M). We will use this curvature form to construct characteristic classes in Section 5.4.

Lemma 3.3. Let f: (M, ω) → (M, ω) × (Σ, ω_{Σ}) be a symplectic embedding given by f (p) = (p, s_{0}), where s_{0} ∈ Σ is a base point on the surface. The image of the curvature 2-form

Θ_{f} : ℋ_{f} × ℋ_{f} → Lie(Symp(M, ω))

contains the subspace of all exact 1-forms.

Proof. What in fact we need to show is that given a function H: M → there are functions F, G: M × Σ → constant on f (M) such that {F, G} ∘ f = H.

Let F′, G′: Σ → be smooth functions such that F′(s_{0}) = 0 and {F′, G′}(s_{0}) = 1. Define F(p, s): = F′(s), G(p, s) = G′(s) and Ĥ(p, s) = H(p). The following calculation follows from standard properties of the Poisson bracket.

{ĤF, G} = Ĥ{F, G} + F{Ĥ, G} = Ĥ{F, G}

Hence, {ĤF, G} ∘ f = Ĥ ∘ f = H. Note that the functions ĤF and G are constant on f (M) as required.

Corollary 3.4. The image of the curvature 2-form

Θ_{i∘f} : ℋ_{i∘f} × ℋ_{i∘f} → Lie(Symp(M, ω))

of the connection in the universal configuration fibration contains the subspace of all exact 1-forms for every f: (M, ω) → ^{∞} .

3.3. The holonomy group. Let (M, ω) →^{i} E →^{π} B be a symplectic fibration. Let K := ker [ π_{∗} : H_{2}(E, ) → H_{2}(B, )]. The following exact sequence:

0 → H_{2}(F, ) → K → H_{1}(B; H_{1}(F, )) → 0

is derived from the Leray-Serre spectral sequence associated with E. An element of H_{1}(B, H_{1}(F, )) may be described as a (class of a) loop γ in B and a γ-invariant cycle [ℓ] ∈ H_{1}(F, )^{γ} in the fiber F over a base point of γ.

Let Ω be a closed extension of ω. We construct certain lift Σ_{γ,ℓ} ∈ H_{2}(E, ) of (γ, ℓ) ∈ H_{1}(B; H_{1}(M, )) to K. It is a union of two chains f (S^{1} × I) and C. The map f is such that f ({s} × I) is a horizontal lift of γ for each s and f (S^{1} × {0}) = ℓ. The chain C is a chain in F such that ∂C = γ_{∗}ℓ − ℓ, where γ_{∗} is the holonomy of γ with respect to Ω. Observe that Ω(f_{∗}(\frac{d}{ds}), f_{∗}(\frac{d}{dt})) = 0 because the kernel of Ω is the horizontal subspace. Thus ∫_{Σ_{γ,ℓ}} Ω = ∫_{C}ω.

Let ψ ∈ Symp(M, ω) and H_{1}(M, )^{ψ} = {ℓ ∈ H_{1}(M, ): ψ_{∗}(ℓ) = ℓ} be the ψinvariant part of the first cohomology. Define a flux-like homomorphism

Flux^{ψ} : H^{1}(M, )^{ψ} → /ω

as follows (here /ω denotes the quotient of by the group of periods of the symplectic form). If [ℓ] ∈ H_{1}(M, )^{ψ} then there exists a 2-chain C such that ∂C = ℓ − ψ_{∗}ℓ. Then

Flux^{ψ}(ℓ) := ∫_{C}ω mod (periods of ω).

The integral is defined up to the period of ω and does not depend on the choice of representative ℓ in the homology class.

Let \overline{𝒟} be a subset of Symp(M, ω̂) consisting of those ψ ∈ Symp(M, ω̂) for which Flux^{ψ} = 0. Note that \overline{𝒟} is conjugacy invariant, but it is not a subgroup of Symp(M, ω̂).

Proposition 3.5. Let M → E → B be a symplectic fibration. Let Ω be an extension of the symplectic form ω and let K := ker H_{2}(E, ) → H_{2}(B, ). Then ⟨Ω, K⟩ ⊂ if and only if the holonomy π_{1}(B) → Symp(M, ω̂) is contained in \overline{𝒟} Proof. If C′ ∈ H_{2}(M, ) then ∫_{i_{∗}(C′)}Ω = ∫′_{C}ω since Ω extends ω. If (γ, ℓ) ∈ H_{1}(B; H_{1}(M, )) then

∫_{Σ_{γ,ℓ}} Ω = ∫_{C}ω = Flux^{ψ}(ℓ) mod (periods of ω),

where ψ is the monodromy around loop γ.

Recall that we define 𝒟 to be the holonomy group of the universal connection θ. Notice that 𝒟 is not well defined as a subgroup of Symp(M, ω) since it depends of the reference point f: (M, ω) → ^{∞} . However, according to standard theory of connections, different holonomy subgroups are conjugate.

Corollary 3.6. The set \overline{𝒟} is the sum of all subgroups conjugated to 𝒟.

Proof. Let E = M × / ∼, where (m, t + k) ∼ (ψ^{k}(m), t) for k ∈ . The form Ω = π_{M}^{∗}ω on M × descends to a closed connection form on E with a holonomy ψ. Since H_{2}(S^{1}, ) = 0 we deduce from 3.5 that if ψ ∈ \overline{𝒟} then Ω has integral periods.

To prove the converse inclusion notice that it is enough to show that 𝒟 ⊂ \overline{𝒟}, according to the conjugacy invariance. Since 𝒟 is the universal holonomy group, for any ψ ∈ 𝒟 there exists a bundle M → E → S^{1} with a closed integral connection form Ω such that ψ is a holonomy of that form. Thus ψ ∈ \overline{𝒟} again by 3.5.

Proposition 3.7.

(1) Let H ⊂ Symp(M, ω̂) be any subgroup. If H ⊂ \overline{𝒟} then H < 𝒟 (up to conjugacy).

(2) 𝒟 ∩ Symp_{0}(M, ω̂) = \overline{𝒟} ∩ Symp_{0}(M, ω̂),

(3) 𝒟 is a closed subgroup of Symp(M, ω̂)

Proof. First we will prove (1) with the additional assumption that H is countable. The we will use it to prove (2) and (3). Finally we will conclude general case of (1) with the use of (3).

(1) We shall construct a fibration over a noncompact surface (possibly of infinite genus) with closed connection form Ω and holonomy equal to H. Since H < \overline{𝒟} then, according to Proposition 3.5 ⟨Ω, K⟩ ⊂ . Because the base is a noncompact surface then the latter is equivalent to the integrality of Ω. To see this notice that K = H_{2}(E, ) as H_{2}(B, ) is trivial. By the universality of 𝒟 we will conclude that, up to a conjugation, H < 𝒟.

Take M_{ψ_{i}} × S^{1} → T^{2} for each ψ_{i} ∈ H, i = 1, 2, … and form a countable fiber connected sum (M_{ψ_{1}} × S^{1})#_{M}(M_{ψ_{2}} × S^{1})#_{M} . . .. It is a bundle over Σ_{∞} the surface of infinite genus and admits a closed connection form. Indeed, every M_{ψ} × S^{1} admits a closed connection form as in the proof of Corollary 3.6. One can easily extend it over the fiber connected sum.

Every loop in the connected sum is a composition of loops in the summands, thus the holonomy of the bundle is contained in H. Since all the generators of H are contained in the holonomy, H equals the holonomy group of that bundle.

(2) Let ψ ∈ \overline{𝒟} ∩ Symp_{0}(M, ω). Observe that Flux^{ψ} : H^{1}(M, ) → / is equal to the value of the usual flux on ψ. This shows that \overline{𝒟} ∩ Symp_{0}(M, ω) ⊂ Flux^{−1}(H^{1}(M, )). In fact, the equality holds, because 𝒟_{0} = Ham(M, ω), according to Theorem 1.7 (2).

Now take one element ψ from each component of \overline{𝒟} ∩ Symp_{0}(M, ω) and generate subgroup H ⊂ Symp_{0}(M, ω). It is clearly countable, hence it is contained in 𝒟 ∩ Symp_{0}(M, ω), due to the part (1). Since 𝒟_{0} = Ham(M, ω) then \overline{𝒟} ∩ Symp_{0}(M, ω) = H · Ham(M, ω) ⊂ 𝒟 ∩ Symp_{0}(M, ω).

(3) Since \overline{𝒟}∩Symp_{0}(M, ω̂) = Flux^{−1}(H^{1}(M, )) is a closed subgroup of Symp_{0}(M, ω) then (3) follows.

To finish the proof we need to show that (1) holds in full generality. Let H′ be a countable dense subgroup of H. Such a group exists because Symp(M, ω) is a separable group modeled on the space of closed 1-forms. Then gH′g^{−1} ⊂ 𝒟 by already proven part of (1) and gHg^{−1} ⊂ 𝒟 by (3).

3.4. Proof of Theorem 1.11. Let τ ∈ H^{2}(M, ) be a preimage of the class [ω] ∈ H^{2}(M, ) of the symplectic form. Let (M, ω) → E → B be a symplectic fibration with the classifying map f : B → B Symp(M, ω̂).

Lemma 3.8. There exists a lift f̂ : B → B (𝒢/ Map_{0}(M, U(1)))if and only if d_{2}(τ) = 0. Here d_{2} is the differential in the spectral sequence associated to E.

Proof. Recall that Map(M, U(1))/ Map_{0}(M, U(1)) = H^{1}(M, ) and we have a fibration

0 → BH^{1}(M, ) → B(𝒢/ Map_{0}(M, (U(1)))) → B Symp(M, ω̂) → 0

induced by the extension 0 → H^{1}(M, ) → 𝒢/ Map_{0}(M, (U(1)) → Symp(M, ω̂) →

0. Observe that the only obstruction to the existence of the lift is a class κ ∈ H^{2}(B, π_{1}(BH^{1}(M, ))) = H^{2}(B, H^{1}(M, )). We shall show that κ vanishes if and only if d_{2}([τ]) vanishes.

Suppose that d_{2}(τ) = 0. This implies that for any map h : Σ → B from a surface the pull-back bundle h^{∗}E → Σ admits an integral extension of τ. Hence there exists a lift { \atop (f ∘ h)}: Σ → B(𝒢/ Map_{0}(M, U(1))). More precisely, the lift is the composition of the lift Σ → B𝒢 (guaranteed by Theorem 1.9 (1⇔3)) and the map induced by the projection 𝒢 → 𝒢/ Map_{0}(M, U(1)). Therefore κ evaluated on h_{∗}[Σ] equals zero. Since h was arbitrary the class κ vanishes.

Now suppose that there exists a lift f̂ : B → B𝒢/ Map_{0}(M, U(1)). The only obstruction for lifting it further to B𝒢 is an element of

H^{3}(B(𝒢/ Map_{0}(M, U(1))); π_{2}(B Map_{0}(M, U(1)))).

Hence it vanishes on two dimensional cycles. This implies that any map h : Σ → B can be lifted to B𝒢. Thus the pull-back bundle admits an integral extension of the class τ (according to Theorem 1.9 (1⇔3)) which means that h^{∗}(d_{2}(τ))= d_{2}(h^{∗}τ) = 0. Since h is arbitrary then we have d_{2}(τ) = 0 over B.

Lemma 3.9 ([8]). If d_{2}(τ) = 0 then d_{3}(τ) ∈ E_{3}^{3,0} is torsion.

Proof. Recall that dim M = 2n and that π_{1}(BSymp(M, ω̂)) acts trivially on H^{2n}(M, ). Then we have that

0 = d_{3}[τ^{n+1}]
= (n + 1)d_{3}[τ] ⊗ [τ^{n}] ∈ H^{3}(BSymp(M, ω); H^{2n}(M, )).

It means that d_{3}[τ] is torsion.

Proof of Theorem 1.11.Assume that there is a lift f̂ : B → B𝒢/ Map_{0}(M, U(1)). Lemma 3.8 and 3.9 show that d_{2}(τ) = 0 and d_{3}(τ) is torsion.

Since all differentials of [ω] vanish, there exists a closed extension Ω′ of ω. We shall define a new closed connection form Ω := Ω′ + π^{∗}β so that it is integral (cf. Section 3 in [11]). Since H_{2}(B, ) is torsion-free then H_{2}(E, ) = π_{∗}(H_{2}(E, )) ⊕ K, where K := ker π_{∗} as in Section 3.3. It follows from Proposition 3.5 that Ω′ has integral periods on the cycles from K. We then define β ∈ Ω^{2}(B) to be any two form satisfying ∫_{π_{∗}(C_{i})} β = − ∫_{C_{i}} Ω′, where {π_{∗}(C_{i})} form a basis of the image π_{∗}(H_{2}(E, )).

Finally, since E admits closed integral connection form it is an integral symplectic configuration, due to Theorem 1.9. Notice that the class of the above integral connection form might not restrict to τ, i.e. τ i^{∗}[Ω] ∈ H^{2}(M, ).

3.5. Other results. We conclude this section with several geometric results. The proofs are straightforward and are left to the reader.

Proposition 3.10.

(1) The action of Symp(W, ω_{W}) on Symp(M, W) given by the composition φ · f: = φ ∘ f preserves the connection θ.

(2) The induced action on the associated bundle (M, ω) → M_{W} → �_{W} is given by φ · [f, p]: = [φ ∘ f, p]. The evaluation map ev: M_{W} → W is Symp(W, ω_{W})- equivariant.

The general idea of this paper is to fix the first variable (M, ω) and investigate the functor �(M, ·) in order to get various informations about (M, ω). One can do the other way around and fix the second variable. This approach was taken by Haller-Vizman in [6].

They write explicitly a moment map for the action and express �_{W} as a coadjoint orbit in the dual Lie algebra of the group Ham(W, ω_{W}).

Proposition 3.11 ([6]).

(1) The fiber integral p_{!}(ev^{∗}(ω_{W}^{n+1})) ∈ Ω^{2}(�_{W}) is a symplectic form on �_{W}.

(2) The action induced on �_{W} preserves the symplectic form

p_{!}(ev^{∗}(ω_{W}^{n+1})).

4. Group cohomology and the group of McDuff

4.1. Preparation on crossed homomorphisms. Let G be a topological group and M be a topological G-module. All the modules we consider are right modules. A continuous map ϕ: G → M is called a continuous crossed homomorphism or continuous 1-cocycle if ϕ(gh) = ϕ(h) + ϕ(g) · h.

Let S be a topological group, R be a topological S-module, and Z its submodule. Let ϕ: S → R/Z be a crossed homomorphism. Define G to be the pull-back in the following diagram:

G −−−⟶^{Φ} R
p↓  ↓
S −−−⟶^{ϕ} R/Z

where vertical arrows are homomorphisms and Φ is a crossed homomorphism with respect to the action of S. In other words G is a subgroup of the semidirect product R S consisting of the pairs (s, r) such that ϕ(s) = r + Z.

Lemma 4.1. Let H be a kernel of ϕ. Then there exist a section σ: H → G (i.e. continuous homomorphism, such that p ∘ σ = id_{H}).

Proof. Consider a pull-back:

G′ −−−⟶^{Φ′} R
p_{H}↓  ↓
H −−−⟶^{ϕ_{|H}} R/Z

One observes that since ϕ_{|H} is trivial then G′ is a semidirect product Z H and, in particular, there is a splitting H → G′. Clearly, G′ is a subgroup of G. Hence composing the splitting H → G′ with the inclusion G′ < G we get the required section.

Note that ker p = Z and consider the two (topological) fibrations:

Z → G →^{p} S,

and

H → S →^{ϕ} R/Z.

Lemma 4.2. The corresponding long exact sequences of homotopy groups are matched up as in the following commutative diagram:

⋯−−−⟶ π_{k+1}(S) −−−⟶ π_{k+1}(R/Z) −−−⟶ π_{k}(H) −−−⟶ ⋯
∥  ^{∂_{R}}↓  ^{π_{k}(σ)}↓
⋯−−−⟶ π_{k+1}(S) −−−⟶ π_{k}(Z) −−−⟶ π_{k}(G) −−−⟶ ⋯

where ∂_{R} is the connecting homomorphism in the long exact sequence of the homotopy groups of the fibration Z → R → R/Z.

Proof. This is straightforward to check using standard diagram-chasing.

Corollary 4.3. If R is contractible then σ constructed above is a homotopy equivalence.

Proof. If R is contractible then ∂_{R} is an isomorphism. Hence, by the five lemma, π_{k}(σ) is also an isomorphism and the statement follows.

4.2. The cocycles and group of McDuff. Let ω be a closed (possibly singular) 2-form on a manifold W. Let SH_{1}(W; ) be space of integral 1-cycles quotient out by boundaries of chains of zero ω-area. There is a split extension

0 → /ω → SH_{1}(W, ) →^{p} H_{1}(W, ) → 0.

The existence of a section s: H_{1}(W, ) → SH_{1}(W, ) follows since /ω is divisible. Precisely such a section may be constructed generator-by-generator. Assume that the section is defined on a submodule V ⊂ H_{1}(W, ), v ∈ H_{1}(W, ) and n is the least integer such that nv ∈ V. Let ṽby any lift of v in SH_{1}(W, ). If n = ∞ we set s(v) = ṽ. In the other case we see that s(nv) − nṽ= [r] ∈ /ω. If γ\frac{r}{n} is a contractible loop bounding a disk of area r/n we set s(v) = ṽ+ γ\frac{r}{n}.

Clearly, any group G that fixes ω acts on SH_{1}(W, ). In [11] McDuff defines a continuous crossed homomorphism F_{s}: G → H^{1}(M; /ω) as follows

F_{s}(g)(γ): = s(g_{∗}(γ)) − g_{∗}(s(γ)) ∈ /ω = ker (SH_{1}(W, ) → H_{1}(W, )) .

Straightforward computation shows that F_{s} satisfies

F_{s}(gh)(γ) = g_{∗}(F_{s}(h)(γ)) + F_{s}(g)(h_{∗}(γ)).

Since G acts trivially on /ω then g_{∗}(F_{s}(h)(γ)) = F_{s}(h)(γ). Hence

F_{s}(gh) = F_{s}(g) + (F_{s}(g)) · h,

that is F_{s} is indeed a crossed-homomorphism.

We shall apply this construction in the following situation:

(1) (W_{1}, ω_{1}) = (L_{τ}, π^{∗}ω) where L_{τ} is a prequantum U(1)-bundle with [L] = τ ∈

H^{2}(M, ) and G_{1} = 𝒢.

(2) (W_{2}, ω_{2}) = (M, ω) and G_{2} = Symp(M, ω̂).

Lemma 4.4.

(1) Since π^{∗}ω is exact on L_{τ} then the group of periods is trivial i.e. H^{1}(L_{τ}, /π^{∗}ω) = H^{1}(M, ).

(2) One can choose sections s̃: H_{1}(L_{τ}, ) → SH_{1}(L_{τ}, ) and s: H_{1}(M, ) → SH_{1}(M, ) such that the following diagram commutes.

\xymatrix{ \ar[r] \ar[d] & SH_{1}(L_{τ}, ) \ar[r] \ar[d] & H_{1}(L_{τ}, ) \ar[d]^{=} \ar[l]_{s̃} \\ /ω \ar[r] & SH_{1}(M, ) \ar[r] & H_{1}(M, ) \ar[l]_{s}}

(3) If ϕ: 𝒢 → Symp(M, ω̂) assigns to a bundle map its underlying symplectomorphism then the following diagram commutes:

𝒢  −−−⟶^{Fs̃} H^{1}(M; )
ϕ ↓  ↓
Symp(M, ω̂) −−−⟶^{F_{s}} H^{1}(M; /ω)

(4) The restriction of Fs̃ to Map_{0}(M, U(1)) is trivial, and thus descends to a crossed homomorphism F̂_{s}: 𝒢/ Map_{0}(M, U(1)) → H^{1}(M; ).

(5) The following diagram

𝒢/ Map_{0}(M, U(1)) \underrightarrow{F̂s̃} H^{1}(M; )
↓  ↓
Symp(M, ω) −−−⟶^{F_{s}} H^{1}(M; / )

is a pull-back.

Proof. Most of the statements are trivial.

(2) First we have to define s̃on K = ker H_{1}(L_{τ}, ) → H_{1}(M, ) in such a way that its image lives in ker SH_{1}(L_{τ}, ) → SH_{1}(M, ). The generator of K is represented by a loop in a fiber, and the value of s̃may be also be taken to be a loop in the fiber.

Then we extend s̃to whole H_{1}(L_{τ}, ) as explained above. It clearly descends to the map s: H_{1}(M, ) → SH_{1}(M, ).

(4) This is obvious since Map_{0}(M, U(1)) is connected and the fiber of the map H^{1}(M, ) → H^{1}(M, / ) is discrete.

Remark 4.5 (about Lemma 4.4 (2)). McDuff [11] shows that for each s one can choose a lift τ of [ω] and a section s̃such that s is constructed as above.

McDuff proves that the restriction of F_{s} to Symp_{0}(M, ω) is equal to the flux homomorphism. She defines Ham^{s} (M, ω): = ker F_{s}. Clearly, the component of the identity of Ham^{s} (M, ω) is equal to Ham(M, ω).

4.3. The groups Ham^{s} and 𝒟 are equal. Let G be a topological group and F: G_{0} → A a G-equivariant homomorphism to a right G-module A. That is it satisfies F(f^{−1} g f) = F(g) · f. Let H < G be a subgroup intersecting every connected component of G and such that H ∩ G_{0} = ker F. Let σ: G/G_{0} = π_{0}(G) → H be a set theoretic section of the natural projection p: G → G/G_{0}. Define 1-cocycle F^{σ}: G → A by

F^{σ}(g): = F(σ(p(g))^{−1}g).

The following lemma that is a reformulation of Proposition 1.8(ii) from [11].

Lemma 4.6. The 1-cocycle F^{σ} extends F and H = ker F^{σ}.

Proof. It is straightforward that F^{σ} extends F. To prove that F^{σ} is a 1-cocycle choose f, g ∈ G. Since σ(G) ⊂ H then h = δ(σ ∘ p)(f, g) = σ(p(f g))σ(p(g))^{−1}σ(p(f))^{−1} ∈ H. We calculate

G_{0} ∋ σ(f g)^{−1}f g = hσ(p(g))^{−1}σ(p(f))^{−1}f g
= hσ(p(g))^{−1}gg^{−1}σ(p(f))^{−1}f g
= h (σ(p(g))^{−1}g)(g^{−1}σ(p(f))^{−1}f g) ∈ hG_{0}.

Thus h ∈ G_{0} and finally h ∈ ker F = G_{0} ∩ H. Then

F^{σ}(f g) = F (σ(p(f g))^{−1}f g)
= F(h) + F (σ(p(g))^{−1}g)+ F (g^{−1}σ(p(f))^{−1}f g)
= F^{σ}(g) + F^{σ}(f) · g

The proof that H = ker F^{σ} is also immediate and is left to the reader.

Theorem 4.7.

(1) There exists a continuous group homomorphism Ham^{s} (M, ω) → 𝒢/ Map_{0}(M, U(1)) that is a homotopy equivalence.

(2) The group Ham^{s} (M, ω) and the holonomy group 𝒟 are isomorphic.

Proof. (1) Since H^{1}(M; ) is contractible then the statement follows from Corollary 4.3.

(2) Step 1. We will show that Ham^{s} (M, ω) ⊂ \overline{𝒟}. We have to observe that if

[l] ∈ H_{1}(M, )^{ψ} then F_{s}(ψ)(l) = Flux^{ψ}(ℓ). Step 2. By Proposition 3.7 (1) we conclude that Ham^{s} (M, ω) ⊂ 𝒟 (up to conjugacy). Step 3. Let F be the following composition

Symp_{0}(M, ω̂) →^{Flux} H^{1}(M, )/Γ_{ω} → H^{1}(M, / ).

From the definition Ham^{s} (M, ω) ∩ Symp_{0}(M, ω̂) = ker F. Let σ be a section π_{0}(Symp(M, ω̂)) → Ham^{s} (M, ω). Then by Lemma 4.6 Ham^{s} (M, ω) = ker F^{σ}. By Theorem 1.7(3) 𝒟 = Ham(M, ω) ∩ Symp_{0}(M, ω̂) = ker F. Since σ(π_{0}(Symp(M, ω̂))) ⊂ Ham^{s} (M, ω) ⊂ 𝒟, by Lemma 4.6 we conclude that 𝒟 = ker F^{σ}.

4.4. An obstruction associated to a crossed homomorphism. Let V be an Smodule, i.e. there is a homomorphism ρ: S → Aut(V) =: A. A crossed homomorphism S → V may be understood as a deformation of ρ. Precisely the map

(ψ, ρ): S → V A = Ã

is a homomorphism if and only if ψ is a crossed homomorphism.

A (homomorphic) section s: A → Ã of the extension

0 → V → Ã →^{p} A → 0

defines an universal cocycle Ψ^{s}: Ã → V by the formula Ψ^{s}(a) = (sp(a))^{−1} a. It is universal in the following sense. Given a homomorphism h: S → Ã one defines a crossed homomorphism ψ_{h}^{s}: S → V by the composition ψ_{h}^{s} = Ψ^{s} ∘ h associated to the action p ∘ h: S → A. Different choice of a section s: A → Ã changes Ψ^{s} by a coboundary.

The kernel H: = ker ψ is a pull-back of the following diagram:

\begin{matrix}H & −−−⟶ & A \\ ↓ & & ↓^{s} \\ S & −−−⟶^{h} & Ã \end{matrix}

Where h = (ψ, ρ). Thus BH is a homotopy pull-back of the following diagram

BH −−−⟶ BA
↓  ↓^{Bs}
BS −−−⟶^{Bh} BÃ

The homotopy kernel of the map BA → BÃ equals to V, moreover the fibration

V → BA = V Ã EÃ → BÃ

is the universal V-bundle over BÃ. Note that V is an affine Ã-space and it is an A-torsor.

In particular the obstruction to the existence of a section BS → BH is a pull-back of the obstruction to the existence of a section BÃ → BA.

In our set-up S = Symp(M, ω̂) and V = H^{1}(M, )/H^{1}(M, ) and, in particular, V = K(H^{1}(M, ), 1). Thus the total obstruction class 𝒪_{V} to the existence of the lift BÃ → BA is the element of H^{2}(BÃ, π_{1}(V)) = H^{2}(BÃ, H^{1}(M, )).

Theorem 4.8 (McDuff). There exists a class 𝒪_{M} ∈ H^{2}(B Symp(M, ω̂), H^{1}(M, )) with the following property. A Symp(M, ω̂)-bundle M → P → B admits a reduction to 𝒟_{τ} if and only if f^{∗}𝒪 = 0 ∈ H^{2}(B, H^{1}(M, )), where f: B → B Symp(M, ω̂) is the classifying map for the bundle P.

Proof. We set 𝒪_{M} = (Bh)^{∗}𝒪_{V} where h = (ρ, F^{s}), ρ is an action of Symp(M, ω̂) on H^{1}(M, ) and F^{s} is the cocycle of McDuff.

In 4.1 we have described the Bockstein map β which, for given short exact sequence of S-modules

0 → Z → R → R/Z → 0,

to every (cohomology class of a) crossed homomorphism S → R/Z associates an extension 0 → Z → G → S → 0. If Z is discrete, then such an extension is classified by an element σ_{G} ∈ H^{2}(BS, Z) (cf. [14]).

Corollary 4.9. 𝒪_{M} = σ_{𝒢/ Map_{0}(M,U(1))} = β([F^{s}]).

Proof. Recall that Ham^{s} and 𝒢/ Map_{0}(M, U(1)) are homotopy equivariant, thus the obstruction to the existence of a section Ham^{s}(M, ω) → Symp(M, ω̂) equals the the obstruction to the existence of a section 𝒢/ Map_{0}(M, U(1)) → Symp(M, ω̂).

The second equality follows directly from the definition of β.

5. Cohomology ring of symplectic configurations

5.1. Characteristic classes of configurations. We define a characteristic class of configurations of (M, ω) in (W, ω_{W}) to be an element of the cohomology ring H^{∗}(�_{W}). If M compact then certain characteristic classes can be obtained as fiber integrals

χ_{k}^{W}: = p_{!}(ev^{∗}([ω_{W}^{n+k}])).

These classes are natural in the sense that if ψ: (W, ω_{W}) → (V, ω_{V}) is a symplectic embedding then χ_{k}^{W} = Ψ^{∗}χ_{k}^{V}, where Ψ: �_{W} → �_{V} is the map induced by ψ.

Since any integral symplectic manifold embeds into ^{N} then we define universal characteristic classes with respect to ^{∞} . We denote the above fiber integrals by χ_{k} ∈ H^{2k}(�). In this case those are usual characteristic classes of a fibration with the structure group 𝒢.

Fundamental question is then if these classes are symplectic that is if they come from H^{∗}(B Symp) via the canonical map � → B Symp. In general the answer is negative.

Lemma 5.1. Let σ ∈ π_{2}(�) be an element coming from H^{0}(M; ) (see Theorem 1.5). Then ⟨χ_{1}, σ⟩ 0. In particular, χ_{1} is not symplectic class. Moreover, χ_{1} is of infinite order in H^{∗}(�).

Proof. Recall that σ is represented by a configuration M × S^{2} → ^{∞} such that the S^{2} summand is represents the positive generator of π_{2}( ^{∞} ). Clearly, ev^{∗}(ω_{∞} ) = ω ⊕ ω_{S^{2}} and hence π_{!}(ev^{∗}(ω_{∞}^{n+1})) = (n + 1) ω_{S^{2}}. Since the configuration is a trivial fibration then no symplectic characteristic class can be nontrivial.

In order to prove the last statement, consider a configuration P: = M × (S^{2} × ⋯ ×S^{2}) → ^{∞} such that every S^{2} is as above. Clearly, χ_{1}(P) is a sum of generators and its top power is nonzero. Since it works for any number of factors S^{2}, then it follows that χ_{1} is of infinite order.

Let (M, ω) →^{i} M_{Ham(M,ω)} →^{p} BHam(M, ω) be the universal Hamiltonian fibration. Recall that the coupling class Ω ∈ H^{2}(M_{Ham(M,ω)}) is defined by the following two conditions:

(1) i^{∗}(Ω) = [ω]

(2) p_{!}(Ω^{n+1}) = 0

One can define characteristic classes by μ_{k}: = p_{!}(Ω^{n+k}) ∈ H^{2k}(BHam(M, ω)) [7, 8].

Let ˜� : = Symp(M, ^{∞} )/ Ham(M, ω) and let (M, ω) →^{i} M˜� →^{p} ˜� be the associated bundle. Notice that ˜� → � is the universal cover. We have the following commutative diagram of classifying spaces:

\xymatrix{˜� \ar[r] \ar[d] & � \ar[d] \\ BHam(M, ω) \ar[r] & BSymp(M, ω)}

We want to compare the pull-backs of characteristic classes χ_{k} and μ_{k} in H^{∗}(˜� ). To avoid clumsy notation we will denote these pull-backs by the same symbols as the original classes. The guiding idea is that the χ_{k} classes might be easier to calculate than μ_{k}’s.

Lemma 5.2. Let (M, ω) →^{i} M˜� →^{p} ˜� be the above associated bundle. Then χ_{k} μ_{k} modulo an ideal in H^{∗}(˜� ) generated by χ_{1}.

Proof. First observe that the coupling class Ω = ev^{∗}(ω_{0}) − \frac{1}{n+1} p^{∗}(χ_{1}). Calculating the appropriate fiber integral we get the statement.

χ_{k} = p_{!}(ev^{∗}(ω_{0}^{n+k}))
= p_{!} ((Ω + \frac{1}{n + 1} p^{∗}(χ_{1}))^{n+k})

= p_{!} (∑_{i=0}^{n+k} ( \begin{matrix}n + k \\ i \end{matrix} )Ω^{n+k−i} (\frac{1}{n + 1} p^{∗}(χ_{1}))^{i} )

= ∑_{i=0}^{n+k} ( \begin{matrix}n + k \\ i \end{matrix} )(\frac{1}{n + 1} )^{i} p_{!}(Ω^{n+k−i}) (χ_{1})^{i}

= ∑_{i=0}^{k} ( \begin{matrix}n + k \\ i \end{matrix} )(\frac{1}{n + 1} )^{i} μ_{k−i} (χ_{1})^{i}

Corollary 5.3. Let (M, ω) → P → B be a configuration whose classifying map admits a lift to ˜� . If H^{2}(B) = 0 then χ_{k}(P) = μ_{k}(P). In particular, if χ_{k}(P) is nonzero then so does μ_{k}(P).

5.2. Calculating χ-classes. It is important to know if there are relations between characteristic classes. If (M, ω) → P → S^{2k} is a symplectic configuration with nontrivial χ_{k}(P) then χ_{k} is not a product in H^{∗}(�). Indeed, if it were a product then its pull-back χ_{k}(P) ∈ H^{2k}(S^{2k}) would vanish.

Theorem 5.4. Let (M, ω) → P → S^{2k} be a symplectic configuration, k > 1. The following conditions are equivalent.

(1) χ_{k}(P) 0;

(2) μ_{k}(P) 0;
(3) ev_{∗}[P] 0 in H_{2n+2k}( ^{∞} ; );

(4) There exists a ∈ H^{2}(P) extending [ω] such that a^{n+k} 0; in particular P is

c-symplectic. Moreover, any of these conditions implies that k ≤ n + 1.

Proof. The first two conditions are equivalent, according to Corollary 5.3. Each of them is equivalent to the third by the basic properties of fiber integration:

⟨χ_{k}(P), [S^{2k}]⟩= ⟨ev^{∗}(ω_{∞}^{n+k}), [P]⟩= ⟨ω_{∞}^{n+k}, ev_{∗}[P]⟩.

The third implies the fourth since the c-symplectic class is given by [ev^{∗}(ω_{∞} )]. The converse implication follows from Theorem 1.9.

For the last statement, observe that H^{2n+2}(P) = 0 if k > n + 1. Hence P cannot be c-symplectic.

Remark 5.5. Consider the question of finding a large k ∈ such that the Hurewicz map π_{2k}(BSymp(M, ω)) → H_{2k}(BSymp(M, ω); ) is nontrivial. The characteristic classes constructed as fiber integrals give quite restricted method for answering this question. As mentioned in the above theorem one can detect homologically nontrivial spheres up to dimension dim M + 2.

5.3. Examples.

(1) Consider the symplectic fibration ^{1} → ^{3} → S^{4} = ^{1}. The total space is symplectic hence, due to Theorem 5.4, we get that μ_{2} = χ_{2} 0 in H^{4}(�(S^{2})).

(2) Moreover, take a symplectic fibration F → P → ^{3} such that P admits a compatible symplectic structure. Composing with the above we get another symplectic fibration

(M, ω) → P → S^{4},

whose fiber is the total space of F → (M, ω) → ^{1}. Again, we obtain that χ_{2} = μ_{2} 0 in H^{4}(�(M, ω)).

(3) Similarly, consider the symplectic fibration

3 → ^{7} → S^{8} = a .

Here, we get that χ_{4} and μ_{4} are nonzero for ^{3}.

(4) Taking a symplectic fibration over ^{7} admitting a compatible symplectic structure we get a result analogous to the one in (2).

(5) Let (M, ω): = U(m)/U(m_{1}) × ⋯ × U(m_{l}), m = m_{1} + ⋯ + m_{l} and m_{1} ≥ m_{2} ≥ ⋯≥ m_{l} be a generalized flag manifold equipped with the homogeneous symplectic form. It is proved in Kȩdra-McDuff [8] that the classes μ_{k} ∈ H^{2k}(BSymp_{0}(M, ω)) are nonzero for 1 < k ≤ m. Moreover, it is shown that they are nontrivial for certain fibrations over spheres, say (M, ω) → P → S^{2k}, with the structure group SU(m). According to Theorem 5.4, P is c-symplectic, however, it is not clear if it admits any symplectic form in general.

As in the previous examples we consider a symplectic fibration F → E → P and compose it with P → S^{2k}. Suppose that this composition is a symplectic fibration with fiber F → Q → M. Then the classes μ_{k}(E) ∈ H^{2k}(BSymp_{0}(Q)) are nonzero.

5.4. Chern-Weil approach. Recall from Section 3.2 that the universal configuration fibration admits a connection with curvature form Θ. It is an ad-invariant two form on the total space Symp(M, ^{∞} ) of the universal principal fibration with values in the Lie algebra of Symp(M, ω) which is identified with closed 1-forms on M. Since the reduced holonomy group is Hamiltonian (Theorem 1.7 (1)) then the curvature form has values in the Lie algebra of Ham(M, ω) which is identified with C_{0}^{∞} (M), the space of functions on M with zero mean value.

Let 𝒫: C_{0}^{∞} (M) → be an invariant polynomial. It means that 𝒫(h_{1}∘ψ, ..., h_{k}∘ψ) = 𝒫(h_{1}, ..., h_{k}), where ψ ∈ Symp(M, ω) and h_{i} ∈ C_{0}^{∞} (M). Then

χ_{[f]}^{𝒫}(X_{1}, Y_{1}, …, X_{k}, Y_{k}): = 𝒫(Θ_{f} (X_{1}^{h}, Y_{1}^{h}), …, Θ_{f} (X_{k}^{h}, Y_{k}^{h}))

defines a 2k-form on the base �. The definition does not depend on the choice of f in the fiber since the polynomial 𝒫 is invariant. It is a standard fact that these forms are closed [10]. We define universal characteristic classes to be cohomology classes of the forms χ^{𝒫} .

The following formula gives a sequence of invariant polynomials.

𝒫_{k}(h_{1}, …, h_{k}): = ∫_{M} h_{1} …h_{k}ω^{n}

The proof of the next lemma is analogous to the proof of Lemma 3.9 in Kȩdra-McDuff [8].

Lemma 5.6. [χ^{𝒫_{k}} ] = const χ_{k} for k > 1.

References

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[2] Y. Eliashberg and N. Mishachev. Introduction to the h-principle, volume 48 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2002.

[3] Robert E. Gompf. A new construction of symplectic manifolds. Ann. of Math. (2), 142(3):527–595, 1995.

[4] Mikhael Gromov. Partial differential relations, volume 9 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1986.

[5] Victor Guillemin, Eugene Lerman, and Shlomo Sternberg. Symplectic fibrations and multiplicity diagrams. Cambridge University Press, Cambridge, 1996.

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[7] Tadeusz Januszkiewicz and Jarek Kȩdra. Characteristic classes of smooth fibrations. preprint, 2003.

[8] Jarek Kȩdra and Dusa McDuff. Homotopy properties of hamiltonian group actions. Geom. Topol., 2004.

[9] Shoshichi Kobayashi and Katsumi Nomizu. Foundations of differential geometry. Vol. I. Wiley Classics Library. John Wiley & Sons Inc., New York, 1996.

[10] Shoshichi Kobayashi and Katsumi Nomizu. Foundations of differential geometry. Vol. II. Wiley Classics Library. John Wiley & Sons Inc., New York, 1996.

[11] Dusa McDuff. Enlarging the hamiltonian group. in preparation, 2004.

[12] Dusa McDuff and Dietmar Salamon. Introduction to symplectic topology. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1995. Oxford Science Publications.

[13] John W. Milnor and John C. Moore. On the structure of Hopf algebras. Ann. of Math. (2), 81:211–264, 1965.

[14] Graeme Segal. Cohomology of topological groups. In Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), pages 377–387. Academic Press, London, 1970.

[15] Edwin H. Spanier. Algebraic topology. McGraw-Hill Book Co., New York, 1966.

[16] Tammo tom Dieck. Transformation groups, volume 8 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1987.

Gal: Mathematical Institute, University of Wroc�aw, pl. Grunwaldzki 2/4, 50-384 Wroc�aw, Poland Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland

E-mail address: sgal@math.uni.wroc.pl
URL: http://www.math.uni.wroc.pl/~sgal

Kȩdra: Institute of Mathematics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland Department of Mathematical Sciences, University of Aberdeen, Meston Building, King’s College, Aberdeen AB24 3UE, Scotland, UK

E-mail address: kedra@maths.abdn.ac.uk
URL: http://www.maths.abdn.ac.uk/~kedra