Bounded symplectic diffeomorphisms and split flux groups

We prove the bounded isometry conjecture of F. Lalonde and L. Polterovich for a special class of closed symplectic manifolds. As a byproduct, it is shown that the flux group of a product of these special symplectic manifold is isomorphic to the direct sum of the flux group of each symplectic manifold.


Introduction
For a closed symplectic manifold (M, ω), the group Ham(M, ω) of Hamiltonian diffeomorphisms carries a norm called the Hofer norm. The group Ham(M, ω) is a normal subgroup of Symp 0 (M, ω), the group of symplectic diffeomorphisms, and the Hofer norm is invariant under conjugation by Symp 0 (M, ω). For a fixed symplectic diffeomorphisms ψ, the map C ψ : Ham(M, ω) − → Ham(M, ω) defined by C ψ (h) = ψ • h • ψ −1 is an isometry with respect to the Hofer norm. In [5] F. Lalonde and L. Polterovich study the isometries of the group of Hamiltonian diffeomorphisms with respect to the Hofer norm. Based on this they call a symplectic diffeomorphism ψ bounded if the Hofer norm of the commutator [ψ, h] remains bounded as h varies in Ham(M, ω). The set of bounded symplectic diffeomorphisms BI 0 (M ) of (M, ω) is a group that contains all Hamiltonian diffeomorphisms.
F. Lalonde and L. Polterovich conjectured that Ham(M, ω) = BI 0 (M, ω) for any closed symplectic manifold (M, ω). This problem is known as the bounded isometry conjecture. In [5] they proved the conjecture when the symplectic manifold is a surface of positive genus or is a product of these surfaces. In [4] F. Lalonde and C. Pestieau proved the conjecture for the product of a closed surface of positive genus and a simply connected manifold. Recently, Z. Han [2] proved the conjecture for the Kodaira-Thurston manifold.
In fact, in [5] F. Lalonde and L. Polterovich proved a stronger result than the bounded isometry conjecture. They proved that if an equivalence class of Symp 0 (M, ω)/Ham(M, ω) has an unbounded symplectic diffeomorphism, then there is a strongly unbounded symplectic diffeomorphism in the same class. This is equivalent to the fact that for any nonzero element v of H 1 (M )/Γ M there is a strongly unbounded symplectic diffeomorphism with flux v. Here Γ M stands for the flux group of (M, ω). For the details, see Section 3. Here we prove this stronger result.
We prove the bounded isometry conjecture for a closed symplectic manifold (M, ω) of dimension 2n satisfying the following two conditions: (a) There are open sets U 1 , . . . , U l ⊂ M such that each U k is symplectomorphic to T 2n \ B(ǫ k ) with the standard symplectic form. Here T 2n is the 2n-dimensional torus and B(ǫ k ) is the embedded image of the standard closed ball in R 2n for a sufficiently small ǫ k ≥ 0. (b) Let j k : U k − → M be the inclusion map and j k * : A symplectic manifold satifying the conditions above is said to satisfy (H). Unless otherwise stated, throughout this article cohomology H * (·) stands for de Rham cohomology and H * c (·) stands for de Rham cohomology with compact support.  The proof of Theorem 1.1 is based on the fact that the bounded isometry conjecture holds for the punctured torus (T 2n * , ω 0 ). This was shown in [5] for n = 1, but in fact their argument works for all n. For the sake of completeness we prove that BI 0 (T 2n * , ω 0 ) = Ham c (T 2n * , ω 0 ) in Proposition 4.7. Here Ham c (M, ω) stands for the group of Hamiltonian diffeomorphisms of (M, ω) with compact support.
The first example of a closed symplectic manifold satisfying (H) is a closed surface (Σ g , ω) with g embedded punctured tori. Another example is the blow-up of the torus (T 2n , ω 0 ) at one point, or more generally the blow-up of (T 2n , ω 0 ) along a simply connected symplectic submanifold. In Section 2, we give some more examples of symplectic manifolds that satisfy (H).
The bounded isometry conjecture holds for a wider class of symplectic manifolds that just those that satisfy (H). Corollary 1.2. Let (M, ω) be a closed symplectic manifold that satisfies (H), and (N, η) a closed symplectic manifold such that H 1 (N ) is trivial or satisfies (H). Then As a consequence of our argument in the proof of the bounded isometry conjecture for this particular class of manifolds, we obtain an interesting result about the flux group. We show that the flux group of a product of two closed symplectic manifolds is isomorphic to the direct sum of the flux group of each manifold. That is, if (M, ω) and (N, η) are symplectic manifolds as in Corollary 1.2 with flux groups Γ M and Γ N , then Γ M×N ≃ Γ M ⊕ Γ N , where Γ M×N is the flux group of (M × N, ω ⊕ η). When this relation holds, we say that the flux group of (M × N, ω ⊕ η) splits.
When (M, ω) is a closed surface of genus greater than one, the flux group is trivial; when g = 1, the flux group equals Z 2 . In [5,Remark 4.3.E], F. Lalonde and L. Polterovich showed that the flux group splits when the manifold is a product of closed surfaces of positive genus. They achieved this in their study of bounded symplectic diffeomorphisms. Here we follow closely their ideas. In Section 2 we give an application of Theorem 1.3 to the fundamental group of Ham(S 2 × Σ g ). Finally we point out an equivalent statement to that of Theorem 1.3. Theorem 1.4. Let (M, ω) and (N, η) be closed symplectic manifolds as in Corollary 2. If ψ ∈ Symp 0 (M, ω) and φ ∈ Symp 0 (N, η) are such that ψ×φ is a Hamiltonian diffeomorphism of (M × N, ω ⊕ η), then ψ and φ are Hamiltonian diffeomorphisms.
Finally we make the remark that all the results remain true in the noncompact case, as long as one considers diffeomorphisms with compact support.
The authors thank Pro. K. Ono for helpful comments on the first draft of this note, and to Prof. D Ruberman and Prof. L. Tu for their valuable comments on improving the exposition of this note. The second author wishes to thank ICTP, Trieste for its hospitality during part of the work on this paper.

Examples
Example. Consider the torus (T 2n , ω 0 ) with its standard symplectic form. Let (T 2n ,ω 0 ) be its blow-up at one point. See [9]. There is a small ǫ > 0 such that the inclusion More generally, let N be a simply connected symplectic submanifold of (T 2n , ω 0 ). Denote by (T 2n N ,ω 0 ) the blow up of (T 2n , ω 0 ) along N . Since N is simply connected, the blow up mapT 2n N ,ω 0 ) by Theomre 1.1. Thus, in every dimension we have new examples of symplectic manifolds that satisfy the bounded isometry conjecture. The next example explores some consequences of Theorem 1.3. Example. Consider the symplectic embedding (T 2n , ω 0 ) − → (T 2(m+n) , ω) in the last 2n coordinates. The symplectic form on the torus is the canonical symplectic form. Thus It is also possible to show directly that (T 2(m+n) \ T 2n , ω) satisfies the bounded isometry conjecture. Our arguments in the proof of Proposition 4.7 apply to this case with no major changes. In fact, condition (H) can be weakened by allowing the set U to be symplectomorphic to T 2(m+n) \ T 2n , and not only symplectomorphic to a punctured torus.

The flux morphism
First a word of warning: if G is a group of diffeomorphisms, we will use ψ to denote an element in π 1 (G) and also to denote a diffeomorphism. It will be clear from the context what it represents.
Let (M, ω) be a closed symplectic manifold and ψ = {ψ t } 0≤t≤1 a loop that represents an element of π 1 (Symp 0 (M, ω)). The isotopy {ψ t } induces a time-dependent vector field X t given by the equation Then the flux morphism Flux M : This map is well defined, that is, it depends only on the homotopy class in Symp 0 (M, ω) based at the identity, and is a group morphism. The image of Flux M is denoted by Γ M and is called the flux group of (M, ω). The rank of Γ M is bounded by the first Betti number b 1 (M ) and is a discrete subgroup of H 1 (M ) (see [7] and [12]). Moreover, the flux morphism fits into the exact sequence of abelian groups where the first map is induced by inclusion and the next one is the flux morphism.
The flux morphism can also be defined on Symp 0 (M, ω), rather than on its fundamental group. In this case for a given symplectic diffeomorphism ψ one considers a symplectic isotopy that joints 1 M with ψ; this will induced a time-dependent vector field X t as before. As in the previous case we have the map Flux M : Symp 0 (M, ω) − → H 1 (M )/Γ M . There is also an exact sequence for this morphism, where the first map is inclusion and the last one is the flux morphism just defined. Note that if ψ and φ are symplectic diffeomorphisms with the same flux, then by the exact sequence (3) there is a Hamiltonian diffeomorphism θ such that ψ = φ • θ. This observation will be used later.
Finally, the flux morphism can also be defined for noncompact symplectic manifolds. In this case one considers symplectic diffeomorphisms with compact support, and the flux morphism takes the form Flux  [11] and of L. Polterovich [14].
Consider two closed symplectic manifolds (M, ω) and (N, η). Then (M × N, ω ⊕ η), where ω ⊕ η stands for π * M (ω) + π * N (η), is also a symplectic manifold. The map Ψ : given by Ψ(ψ, φ) = ψ×φ is a well-defined group homomorphism. It also follows that Γ M ⊕Γ N is a subgroup of Γ M×N , so the induced map i 0 : Here the horizontal maps are Flux M ⊕ Flux N and Flux M×N . And the vertical maps are Ψ and i 0 . So defined, the diagram commutes and the lemma follows.
This lemma is the link between the theory of bounded symplectic diffeomorphisms and our question about the splitting of the flux group. Recall that the universal cover (M ,ω) of (M, ω) is also a symplectic manifold; moreover the projection map π :M − → M satisfies π * (ω) =ω. Let ψ be a Hamiltonian diffeomorphism of (M, ω) and H t : M − → R a Hamiltonian function, whose time-one flow is ψ. Then H t • π generates a Hamiltonian flow on (M ,ω), with time-one mapψ. So definedψ is a Hamiltonian diffeomorphism that lifts ψ. According to Z. Han [1, Lemma 2.1], every Hamiltonian diffeomorphism has a unique lift to (M ,ω).

Bounded symplectic diffeomorphisms
The concepts of strongly unbounded symplectic diffeomorphisms and lifts of Hamiltonian diffeomorphisms are fundamental in the proof of the bounded isometry conjecture. The reason is that using them one can get large lower bounds for the Hofer norm. By the energy-capacity inequality, if ψ is a Hamiltonian diffeomorphism such that ψ( Here e(A) is the displacement energy of A and c G (A) Gromov's capacity of A ( see [6]). However, this inequality is not enough for closed symplectic manifolds, since the capacity c G (·) is bounded from above. Hence we need to pass to the universal cover of the symplectic manifold, since on this open symplectic manifold there are subsets with arbitrary large capacity.
We will show that the bounded isometry conjecture holds for T 2n \ B(ǫ 0 ). In dimension two this was proved by F. Lalonde and L. Polterovich in [5]. Our proof is just an extension of their arguments.
We review a couple of facts of [5] that we need in the proof of the next proposition. In order to have a clear exposition of the arguments, instead of considering diffeomorphisms of T 2n \ B(ǫ 0 ), we will consider periodic diffeomorphisms of R 2n minus a small ball centered at every point of Z 2n . So let a be a small positive number greater than ǫ 0 ; and for each (k 1 , l 1 , . . . , k n , l n ) ∈ Z 2n , consider the small box {(x 1 , y 1 , . . . , x n , y n ) : |x j − k j | ≤ 3a and |y j − l j | ≤ 3a}. Denote by W the union of all such boxes as the point (k 1 , l 1 , . . . , k n , l n ) varies in Z 2n . Let p : R − → R be any smooth 1-periodic function such that Here ǫ is a positive number so small that the definition of p make sense. Finally we also require that  Proof. Since H 1 c (T 2n \ B(ǫ 0 )) = R 2n we can find generators e 1 , f 1 , . . . , e n , f n of H 1 c (T 2n \ B(ǫ 0 )) that are dual to the canonical cycles of the torus. Also let (a 1 , b 1 , . . . , a n , b n ) be a 2n-tuple of non negative real numbers not all of which are zero. We will define ψ i , φ i ∈ Symp c 0 (T 2n \ B(ǫ 0 )) with flux a i e i and b i f i respectively. Let a and W as above. Consider a smooth 1-periodic function h aj : R − → R such that it is equal to zero on [0, 1/3] and [2/3, 1], is positive otherwise, and satisfies a + Similarly for each j we have a function h bj satistying the same properties with b j instead of a j . Then define the symplectic diffeomorphisms ψ j and φ j of (R 2n , ω 0 ) as ψ j (x 1 , y 1 , . . . , x n , y n ) = (x 1 , . . . , x j , y j + a + h aj (x j ), . . . , y n ) and φ j (x 1 , y 1 , . . . , x n , y n ) = (x 1 , . . . , x j + a + h bj (y j ), y j , . . . , y n ) outside W and fix the points close to each B(ǫ 0 ). The maps ψ j correspond to the time-one map of the symplectic flow (x 1 , y 1 , . . . , x n , y n ) → (x 1 , . . . , x j , y j + t(a + h aj (x j )), . . . , y n ) and similarly for φ j . Basically the maps ψ j and φ j are translations along the y j -axes and x j -axes of R 2n respectively. From Equation (4) it follows that Flux(ψ j ) = a j and Flux(φ j ) = b j . Then the flux of ψ 1 • φ 1 • · · · • ψ n • φ n is equal to a 1 e 1 + b 1 f 1 + · · · + a n e n + b n f n . Recall a j and b j are assumed to be non negative. If a j is zero, we define ψ j to be the identity diffeomorphism. If a j is negative, we proceed as above with −a j instead of a j and then Flux(ψ −1 j ) = a j We claim that the symplectic diffeomorphism θ = ψ 1 • φ 1 • · · · • ψ n • φ n is strongly unbounded. To see this, consider the symplectic isotopy f t (x 1 , y 1 , . . . , x n , y n ) = (x 1 , y 1 + tp(x 1 ), . . . , x n , y n ) where p : R − → R is the 1-periodic function defined above. Since p vanishes on [0, 4a − 2ǫ] and [5a + 2ǫ, 1], each f t leaves W fixed pointwise. The zero mean condition on p, implies that {f t } is a Hamiltonian isotopy. Since f t commutes with φ 1 , ψ 2 , . . . , ψ n and φ n but not with ψ 1 , we have [θ, is the identity on the last 2n − 2 coordinates of R 2n , and in then (x 1 , y 1 )-plane it corresponds to the symplectic diffeomorphism g t constructed in [5], that is, Recall from [5] that g t disjoins a rectangle B t whose area is a function of t. Therefore [θ, f t ](B t × R 2n−2 ) ∩ (B t × R 2n−2 ) = ∅. In R 2 the rectangle B t is symplectomorphic to a disk of the same area. Since the area of B t goes to infinity as t goes to infinity, by the energy-capacity inequality the Hofer norm of [θ, f t ] = [ψ 1 , f t ] goes to infinity as t goes to infinity. Hence θ ∈ Symp c (T 2n \ B(ǫ 0 ), ω) is strongly unbounded.
Remark. It is important to note from the proof of Proposition 4.7 that the (x 1 , y 1 )plane of R 2n and the Hamiltonian isotopy {f t } are not related at all to v ∈ H 1 c (T 2n \ B(ǫ 0 ))/Γ T 2n \B(ǫ0) . This observation will be useful when we generalize this result to closed symplectic manifolds that satisfy hypothesis (H).
Before we extend the previous result to symplectic manifolds that satisfy (H) we need the following lemma. It will be used in order to show that the strongly unbounded diffeomorphism θ defined in proof of Proposition 4.7, would remain strongly unbounded on (M, ω) and not only on the open manifold T 2n \ B(ǫ 0 ). Proof. Since H 1 (M ) is nontrivial, there is an embedding i : R − →M . Moreover since R is contractible, the normal bundle ν is isomorphic to R 2n−1 × R. Put the canonical symplectic for on ν. Then there is symplectic diffeomorphims between a neighborhood of the zero section of ν and a neighborhood of i(R) inM . It follows that (0, ǫ) × R embeds symplectically intoM .
Since a symplectic diffeomorphism with compact support in U can be thought of as a symplectic diffeomorphism on M , there is a natural map τ : Symp c 0 (U, ω) − → Symp 0 (M, ω).

This gives rise to the commutative diagram Flux
Then from the commutative diagram and Proposition 4.7 we have the following result. Consider ψ as a symplectic diffeomorphism in Symp 0 (M, ω). Thus ψ has flux v. It only remains to show that ψ is a strongly unbounded diffeomorphism of (M, ω). Note that ψ is not necessarily strongly unbounded on (M, ω) since ψ U ≥ ψ M .
By Lemma 4.8, we have a symplectic embedding of (a, b) × R intoM , where (a, b) is a small interval. Recall that the symplectic diffeomorphisms ψ is the one from the proof of Proposition 4.7, except that now we consider it on (M, ω). Thus on (M ,ω) we have the same symplectic displacement as before. Hence the same arguments of the proof of Proposition 4.7 apply in this case. Thus ψ is strongly unbounded in Symp 0 (M, ω).
Remark. From this result it follows that BI 0 (Σ g , ω) = Ham(Σ g , ω), for g ≥ 1. The argument presented here is different from the proof that appears in [5]. But still the heart of our argument is the same as their approach, namely Proposition 4.7.
For completeness we recall the following result that we will need later. It corresponds to Lemma 4.2 from [4]. Proof. Let c be a positive real number. Since ψ is strongly unbounded there is a Hamiltonian diffeomorphism h of (M, ω) such that the lift of [ψ, h] toM disjoins a ball B 2n (c 0 ) of capacity Thus by the stable version of the energy-capacity inequality of F. Lalonde and C. Pestieau  Theorem 4.11. Let (M, ω) be a symplectic manifold that satisfies hypothesis (H), and let (N, η) be any closed symplectic manifold. If ψ × φ ∈ Symp 0 (M × N ) is a bounded symplectic diffeomorphism, then ψ is a Hamiltonian diffeomorphism of (M, ω).
Proof. Assume that ψ is not a Hamiltonian diffeomorphism. By Theorem 1.1 we have BI 0 (M, ω) = Ham(M, ω), so ψ is an unbounded symplectic diffeomorphisms. Let v ∈ H 1 (M )/Γ M be the flux of ψ. Since v is nonzero, it follows from Proposition 4.9 that there is ψ 0 ∈ Symp 0 (M, ω) that is strongly unbounded and has flux equal to v.

Proof of the main results
The proof of Theorem 1.1 follows from Proposition 4.9 where we showed that for any v in H 1 (M )/Γ M there is an unbounded symplectic diffeomorphism with flux v.