scubo.dvi CUBO A Mathematical Journal Vol.12, No¯ 03, (49–69). October 2010 On The Group of Strong Symplectic Homeomorphisms AUGUSTIN BANYAGA Department of Mathematics, The Pennsylvania State University, University Park, PA 16802 email: banyaga@math.psu.edu ABSTRACT We generalize the “hamiltonian topology” on hamiltonian isotopies to an intrinsic “symplec- tic topology” on the space of symplectic isotopies. We use it to define the group SS ym peo (M,ω) of strong symplectic homeomorphisms, which generalizes the group Hameo(M,ω) of hamiltonian homeomorphisms introduced by Oh and Müller. The group SS ym peo(M,ω) is arcwise connected, is contained in the identity component of S ym peo(M,ω); it con- tains Hameo(M,ω) as a normal subgroup and coincides with it when M is simply con- nected. Finally its commutator subgroup [SS ym peo(M,ω), SS ym peo(M,ω)] is contained in Hameo(M,ω). RESUMEN Generalizamos la “topología hamiltoniano” sobre isotopias hamiltonianas para una “to- pología simpléctica” intrinseca en el espacio de isotopias simplécticas. Nosotros usamos esto para definir el grupo SS ym peo(M,ω) de homeomorfismos simplécticos fuertes, el qual generaliza el grupo Hameo(M,ω) de homeomorfismos hamiltonianos introducido por Oh y Müller. El grupo SS ym peo(M,ω) es conexo por arcos, es contenido en la componente identidad de S ym peo(H,ω); este contiene Hameo(M,ω) como un subgrupo normal y co- incide con este cuando M es simplemente conexa. Finalmente su subgrupo conmutador [SS ym peo(M,ω), SS ym peo(M,ω)] es contenido en Hameo(M,ω). 50 Augustin Banyaga CUBO 12, 3 (2010) Key words and phrases: Hamiltonian homeomorphisms, hamiltonian topology, symplectic topology, stromg symplectic homeomorphisms, C0 symplectic topology. Math. Subj. Class.: MSC2000:53D05; 53D35. 1 Introduction No natural metric on the group S ym p(M,ω) of symplectic diffeomorphisms of a symplectic manifold (M,ω) is known. In this paper we construct a “Hofer-like” metric, depending on several ingredients. However, we prove that all these metrics are equivalent and hence define a natural metric topology on S ym p(M,ω) ( theorem 1’). We use this natural topology on S ym p(M,ω) to define a new group of symplectic homeomorphisms, herein called the group of strong symplectic homeomorphisms (Theorem 2). This group may carry a Calabi invariant. The Eliashberg-Gromov symplectic rigidity theorem says that the group S ym p(M,ω) of symplectomorphisms of a closed symplectic manifold (M,ω) is C0 closed in the group Diff ∞(M) of C∞ diffeomorphisms of M [7],[9]. This means that the “symplectic” nature of a sequence of symplectomorphisms survives topological limits. Also Lalonde-McDuff-Polterovich have shown in [11] that for a symplectomorphism, being “hamiltonian” is topological in nature. These phenomenons attest that there is a C0 symplectic topology underlying the symplectic geometry of a closed symplectic manifold (M,ω). According to Oh-Müller ([13]), the automorphism group of the C0 symplectic topology is the closure of the group S ym p(M,ω) in the group H omeo(M) of homeomorphisms of M endowed with the C0 topology. That group, denoted S ym peo(M,ω) has been called the group of symplectic homeomorphisms: S ym peo(M,ω) =: S ym p(M,ω). The C0 topology on H omeo(M) coincides with the metric topology coming from the metric d( g, h) = max(su px∈M d0( g(x), h(x)), su px∈M d0( g −1 (x), h −1 (x)) where d0 is a distance on M induced by some riemannian metric [3]. On the space P H omeo(M) of continuous paths γ : [0, 1] → H omeo(M), one has the dis- tance d(γ,µ) = su pt∈[0,1] d(γ(t),µ(t)). Consider the space P H am(M) of all isotopies ΦH = [t 7→ Φ t H ] where Φt H is the family of hamiltonian diffeomorphisms obtained by integration of the family of vector fields X H for a CUBO 12, 3 (2010) On The Group of Strong Symplectic Homeomorphisms 51 smooth family H(x, t) of real functions on M, i.e. d dt Φ t H (x) = X H (Φ t H (x)) and Φ0 H = id. Recall that X H is uniquely defined by the equation i(X H )ω = dH where i(.) is the interior product. The set of time one maps of all hamiltonian isotopies {Φt H } form a group, denoted H am(M,ω) and called the group of hamiltonian diffeomorphisms. Definition: The hamiltonian topology [13] on P H am(M) is the metric topology defined by the distance dham(ΦH ,ΦH′ ) = ||H − H ′ ||+ d(ΦH ,ΦH′ ) where ||H − H ′ || = ˆ 1 0 osc(H − H ′ )dt. and the oscillation of a function u is osc(u) = maxx∈M u(x) − minx∈M u(x). Let H ameo(M,ω) denote the space of all homeomorphisms h such that there exists a continuous path λ ∈ P H omeo(M) such that λ(0) = id, λ(1) = h and there exists a Cauchy sequence (for the dham distance) of hamiltonian isotopies ΦHn , which C 0 converges to λ ( in the d metric). The following is the first important theorem in the C0 symplectic topology [13]: Theorem (Oh-Müller): The set H ameo(M,ω) is a topological group. It is a normal sub- group of the identity component S ym peo0 (M,ω) in S ym peo(M,ω). If H 1(M,R) 6= 0, then H ameo(M,ω) is strictly contained in S ym peo0 (M,ω). Remark: It is still unknown in general if the inclusion H ameo(M,ω) ⊂ S ym peo0 (M,ω) is strict. The group H ameo(M,ω) is the topological analogue of the group H am(M,ω) of hamilto- nian diffeomorphisms. 52 Augustin Banyaga CUBO 12, 3 (2010) The goal of this paper is to construct a subgroup of S ym peo0 (M,ω), denoted SS ym peo(M,ω) and nicknamed the group of strong symplectic homeomorphisms, contain- ing H ameo(M,ω), that is: H ameo(M,ω) ⊂ SS ym peo(M,ω) ⊂ S ym peo0 (M,ω). Like H ameo(M,ω), the group SS ym peo(M,ω) is defined using a blend of the C0 topology and the Hofer topology on the space I so(M,ω) of symplectic isotopies of (M,ω). We believe that SS ym peo(M,ω) is “more right” than the group S ym peo(M,ω) for the C0 symplectic topology. In particular the flux homomorphism seems to exist on SS ym peo(M,ω). This will be the object of a future paper. The results of this paper have been announced in [1]. The C0 counter part of the C∞ contact topology is been worked out in [5], [6]. 2 The Symplectic Topology on I so(M,ω) Let I so(M,ω) denote the space of symplectic isotopies of a closed symplectic manifold (M,ω). Recall that a symplectic isotopy is a smooth map Φ : M × [0, 1] → M such that for all t ∈ [0, 1], φt : M → M, x 7→ Φ(x, t) is a symplectic diffeomorphism and φ0 = id. The “Lie algebra” of S ym p(M,ω) is the space s ym p(M,ω) of symplectic vector fields, i.e the set of vector fields X such that i X ω is a closed form. Let φt be a symplectic isotopy, then φ̇t(x) = dφt dt (φ −1 t (x)) is a smooth family of symplectic vector fields. By the theorem of existence and uniqueness of solutions of ODE’s, Φ ∈ I so(M,ω) 7→ φ̇t is a 1-1 correspondence between I so(M,ω) and the space C∞([0, 1], s ym p(M,ω)) of smooth families of symplectic vector fields. Hence any distance on C∞([0, 1], s ym p(M,ω)) gives rise to a distance on I so(M,ω). An intrinsic topology on the space of symplectic vector fields. We define a norm ||.|| on s ym p(M,ω) as follows: first we fix a riemannian metric g (which may be the one we used to define d0 above, or any other riemannian metric), and a basis B = {h1, .., hk } of harmonic 1-forms. For Hodge theory, we refer to [14]. CUBO 12, 3 (2010) On The Group of Strong Symplectic Homeomorphisms 53 Recall that the space harm1 (M, g) of harmonic 1-forms is a finite dimensional vector space over R and its dimension is the first Betti number of M. On harm1 (M, g), we put the following “Euclidean” norm: for H ∈ harm1 (M, g) , H = ∑ λi hi , define: |H |B := ∑ |λi|. This norm is equivalent to any other norm since harm1 (M, g) is a finite dimensional vector space. Here we choose this one for convenience in the calculations and estimates to come later. Given X ∈ s ym p(M,ω), we consider the Hodge decomposition of i X ω [14] : there is a unique harmonic 1-form H X and a unique function u X such that i X ω = H X + du X Recall that the function u X is given by the following formula: u X = δG(i(X )ω), where δ is the codifferential and G is the Green operator (see [14]). This defines a decompsition of X ∈ s ym p(M,ω) as : X = #H X + X u X ,where #H X is de- fined by the equation i(#H X )ω = H X and X u X is the hamiltonian vector field with u X as hamiltonnian. We now define a norm ||.|| on the the vector space s ym p(M,ω) by: ||X|| = |H X |B + osc(u X ). (1) It is easy to see that this is a norm. Let us just verify that ||X|| = 0 implies that X = 0. Indeed |H X |B = 0 implies that i X ω = du X , and osc(u X ) = 0 implies that u X is a constant, therefore du X = 0. Remark: This norm is not invariant by S ym p(M,ω). Hence it does not define a Finsler metric on S ym p(M,ω). The norm ||.|| defined above depends of course on the riemannian metric g and the basis B of harmonic 1-forms. However, we have the following: Theorem 1: All the norms ||.|| defined by equation (1) using different riemannian metrics and different basis of harmonic 1-forms are equivalent. Hence the topology on the space s ym p(M,ω) of symplectic vector fields defined by the norm (1) is intrinsic : it is independent of the choice of the riemannian metric g and of the basis B of harmonic 1-forms. 54 Augustin Banyaga CUBO 12, 3 (2010) For each symplectic isotopy Φ = (φt), consider the Hodge decomposition of i(φ̇t )ω i(φ̇t)ω = H Φ t + du Φ t where H Φt is a harmonic 1-form. We define the length l(Φ) of the isotopy Φ = (φt) by: l(Φ) = ˆ 1 0 (|H Φ t |+ osc(u Φ t ))dt = ˆ 1 0 ||φ̇t||dt. One also writes ˆ 1 0 ||φ̇t||dt = |||φ̇t|||. In the expressions above, we have written |H Φt | for |H Φ t |B , where B is a fixed basis of harm1 (M, g), for a fixed riemannian metric g. We define the distance D0(Φ,Ψ) between two symplectic isotopies Φ = (φt) and Ψ = (ψt) by: D0(Φ,Ψ) = |||φ̇t −ψ̇t||| := ˆ 1 0 (|H Φ t − H Ψ t |+ osc(u Φt − u Ψt ))dt. Denote by Φ−1 = (φ−1t ) and by Ψ −1 = (ψ−1t ) the Inverse isotopies. Remarks: 1. The distance D0(Φ,Ψ) 6= l(Ψ −1 Φ) unless Ψ and Φ are hamiltonian isotopies ( see propo- sition 1). 2. l(Φ) 6= l(Φ−1) unless Φ is hamiltonian. Indeed, H Φ −1 t = −H Φ t but u Φ t is very differerent from uΦ −1 t . The formula of the difference u Φ t − u Φ −1 t follows from propositions 3, 4 and 5. In view of the remarks above, we define a more “symmetrical” distance D by: D(Φ,Ψ) = (D0(Φ,Ψ) + D0(Φ −1 ,Ψ −1 ))/2 Following [13], we define the symplectic distance on I so(M,ω) by: ds ym p (Φ,Ψ) = d(Φ,Ψ) + D(Φ,Ψ). Definition: The symplectic topology on I so(M,ω) is the metric topology defined by the dis- tance ds ym p. Theorem 1’: The symplectic topology on I so(M,ω) is canonical: it is independent of all choices involved in its definition. CUBO 12, 3 (2010) On The Group of Strong Symplectic Homeomorphisms 55 We may also define another distance D∞ on I so(M,ω): D ∞ 0 (Φ,Ψ) = su pt∈[0,1] (|H Φ t − H Ψ t |) + su pt∈[0,1] osc(u Φt − u Ψt )) D ∞ (Φ,Ψ) = (D ∞ 0 (Φ,Ψ) + D ∞ 0 (Φ −1 ,Ψ −1 ))/2 and d ∞ s ym p (Φ,Ψ) = d(Φ,Ψ) + D ∞ (Φ,Ψ) Proposition 1: Let Φ = (φt),Ψ = (ψt) be two hamiltonian isotopies and σt = (ψt) −1φt then |||σ̇t||| = |||φ̇t −ψ̇t||| = ˆ 1 0 osc(u Φ t − u Ψt )dt Proof: This follows immediately from the equation σ̇t = (ψt −1 )∗(φ̇t −ψ̇t), which is a consequence of proposition 4 stated in section 4. Corollary: The distance ds ym reduces to the hamiltonian distance dham when Φ and Ψ are hamiltonian isotopies. The symplectic topology reduces to the “hamiltonian topology” of [13] on paths in H am(M,ω). A Hofer-like metric on S ym p(M,ω)0 For any φ ∈ S ym p(M,ω), define: e0(φ) = in f (l(Φ)) where the infimum is taken over all symplectic isotopies Φ from φ to the identity. The follow- ing result was proved in [2]. Theorem: The map e : S ym p(M,ω)0 → R∪ {∞} : e(φ) =: (e0(φ) + e0(φ −1 ))/2 is a metric on the identity component S ym p(M,ω)0 in the group S ym p(M,ω), i.e. it satisfies (i) e(φ) ≥ 0 and e(φ) = 0 iff φ is the identity. (ii) e(φ) = e((φ)−1) (iii) e(φ.ψ) ≤ (eφ) + e(ψ). The restriction to H am(M,ω) is bounded from above by the Hofer norm. 56 Augustin Banyaga CUBO 12, 3 (2010) Recall that the Hofer norm [10] of a hamiltonian diffeomorphism φ is ||φ||H = in f (l(ΦH )) where the infimum is taken over all hamiltonian isotopies from φ to the identity. The Hofer-like metric above depends on the choice of a riemannian metric g and a basis B of harmonic 1-forms. Hence it is not “natural”. However, by theorem 1, all the metrics constructed that way are equivalent; so they define a natural topology on S ym p(M,ω)0 . 3 Strong Symplectic Homeomorphisms Definition: A homeomorphism h is said to be a strong symplectic homeomorphism if there exists a continuous path λ : [0, 1] → H omeo(M) such that λ(0) = id; λ(1) = h and a sequence Φ n = (φn t ) of symplectic isotopies, which converges to λ in the C0 topology (induced by the norm d) and such that Φn is Cauchy for the metric ds ym p . We will denote by SS ym peo(M,ω) the set of all strong symplectic homeomorphisms. This set is well defined independently of any riemannian metric or any basis of harmonic 1-forms. Clearly, if M is simply connected, the set SS ym peo(M,ω) coincides with the group H ameo(M,ω). We denote by SS ym peo(M,ω)∞ the set defined like in SS ym peo(M,ω) but replacing the norm ds ym p by the norm d ∞ s ym p. Let P H omeo(M) be the set of continuous paths γ : [0, 1] → H omeo(M) such that γ(0) = id, and let P ∞(H arm1 (M) be the space of smooth paths of harmonic 1-forms. We have the following maps: A1 : I so(M,ω) → P H omeo(M),Φ 7→ Φ(t) A2 : I so(M,ω) → P ∞(H arm1 (M),Φ 7→ H Φt A3 : I so(M,ω) → C ∞(M × [0, 1],R),Φ 7→ uΦ Let Q be the image of the mapping A = A1 × A2 × A3 and Q the closure of Q inside I (M,ω) =: P H omeo(M) × P ∞(H arm1 (M) × C∞(M × [0, 1],R), with the symplectic topology, which is the C0 topology on the first factor and the metric topology from D on the second and third factor. Then SS ym peo(M,ω) is just the image of the evaluation map of the path at t= 1 of the image of the projection of Q on the first factor. This defines a surjective map: a : Q → SS ym peo(M,ω) CUBO 12, 3 (2010) On The Group of Strong Symplectic Homeomorphisms 57 The symplectic topology on SS ym peo(M,ω) is the quotient topology induced by a. Our main results are : Theorem 2: The set Q is a topological group. Theorem 3: Let (M,ω) be a closed symplectic manifold. Then SS ym peo(M,ω) is an arc- wise connected topological group (with the sympectic topology), containing H ameo(M,ω) as a normal subgroup, and contained in the path component of the identity S ym peo0 (M,ω) of S ym peo(M,ω). If M is simply connected, SS ym peo(M,ω) = H ameo(M,ω). Finally, the commutator sub- group [SS ym peo(M,ω), SS ym peo(M,ω)] of SS ym peo(M,ω) is contained in H ameo(M,ω). Conjectures: 1. Let (M,ω) be a closed symplectic manifold, then [SS ym peo(M,ω), SS ym peo(M,ω)] = H ameo(M,ω). 2. The inclusion SS ym peo(M,ω) ⊂ S ym peo0 (M,ω) is strict. 3. The results in theorem 3 hold for SS ym peo(M,ω)∞ . Conjecture 3 is supported by a result of Muller asserting that H ameo(M,ω) coincides with H ameo(M,ω)∞ which is defined by replacing the L(1,∞) Hofer norm by the L∞ norm [12]. Measure preserving homeomorphisms On a symplectic 2n dimensional manifold (M,ω), we consider the measure µω defined by the Liouville volume ωn. Let H omeo µω 0 (M) be the identity component in the group of homeomorphisms preserving µω. We have: S ym peo0 (M,ω) ⊂ H omeo µω 0 (M). Oh and Müller [13] have observed that H ameo(M,ω) is a sub-group of the kernel of Fathi’s mass-flow homomorphism [8]. This is a homomorphism θ : H omeo µω 0 (M) → H1(M,R)/Γ, where Γ is some sub-group of H1(M,R). Fathi proved that if the dimension of M is bigger than 2, then K erθ is a simple group. This leaves open the following question [13]: Is H omeo µω 0 (S2) = S ym peo0 (S 2,ω) a simple group? But S ym peo0 (S 2,ω) contains H ameo(S2 ,ω) as a normal subgroup. The question is to decide if the inclusion H ameo(S 2 ,ω) ⊂ S ym peo0 (S 2 ,ω) 58 Augustin Banyaga CUBO 12, 3 (2010) is strict. Since SS ym peo(S2 ,ω) = H ameo(S2 ,ω), our conjecture 2 implies that H omeo µω 0 (S2) = S ym peo0 (S 2,ω) is not a simple group, a conjecture of [13]. Questions 1. Is SS ym peo(M,ω) a normal subgroup of S ym peo0 (M,ω)? 2. Is [S ym peo0 (M,ω), S ym peo0 (M,ω)] contained in H ameo(M,ω)? 4 Proofs of the Results 4.1 Proof of theorem 1 If B and B′ are two basis of harm1 (M, g), then elementary linear algebra shows that |.|B and |.|B′ are equivalent. This implies that the corresponding norms on s ym p(M,ω) are also equivalent. Let us now start our construction with a riemannian metric g and a basis B = (h1, ..hk ) of harm1 (M, g). We saw that for any X ∈ s ym p(M,ω), i X ω = H X + du X and we wrote H X = ∑ λi hi . Let g′ be another riemannian metric. The g′-Hodge decomposition of i X ω is: i X ω = H ′ X + du ′ X where H ′ X is g′-harmonic. Consider the g′-Hodge decompositions of the members hi of the basis B i.e. hi = h ′ i + dvi where h′ i is g′ harmonic. B′ = (h′ 1 , ..h′ k ) is a basis of harm1 (M, g′). Indeed suppose that ∑ r i h ′ i = 0 . The 1-form ∑ r i hi = d( ∑ r i vi) is g-harmonic and exact : ∑ r i hi = d( ∑ r i vi). But an exact harmonic form must be identically zero. Therefore all r i are zero since {hi } form a basis. Hence {h′ i } are linearly independent. The 1-form H ′′ X =: ∑ λi h ′ i is a g′- harmonic form representing the cohomology class of i X ω. By uniqueness, H ′ X = H ′′ X . Hence |H ′ X |B′ = ∑ |λi| = |H X |B CUBO 12, 3 (2010) On The Group of Strong Symplectic Homeomorphisms 59 Furthermore H ′ X = ∑ λi (hi − dvi ) = H X + dv where v = − ∑ λi vi. Hence i X ω = H ′ X + du ′ X = H X + d(v + u ′ X ) By uniqueness in the g-Hodge decomposition of i X ω, u X = v + u ′ X . Denote by ||X||g′ , resp. ||X||g , the norm of X using the riemannian metric g ′ and the basis B ′, resp. using the riemannian metric g and the basis B. Then: ||X||g′ = |H ′ X |B′ + osc(u ′ X ) = |H ′ X |B′ + osc(u X − v) ≤ |H ′ X |B′ + osc(u X ) + osc(−v) = |H X |B + osc(u X ) + osc(v) = ||X||g + osc(v). Let c = 2maxi|vi|, since v = − ∑ λi vi, we get the following inequality: osc(v) ≤ 2max(|v|) ≤ c|H X |B = c|H ′ X |B′ Therefore ||X||g′ ≤ ||X||g + osc(v) ≤ ||X||g + c|H X |B ≤ ||X||g + c(|H X |B + osc(u X )) = (c + 1)||X||g Similarly, ||X||g =|H X |B + osc(u X ) = |H X |B + osc(u ′ X + v) ≤ |H X |B + osc(u ′ X ) + osc(v) =|H ′ X |B′ + osc(u ′ X ) + osc(v) = ||X||g′ + osc(v) ≤ ||X||g′ + c|H ′ X |B′ ≤ ||X||g′ + c(|H ′ X |B′ + osc(u ′ X ) = (c + 1)||X||g′ Hence the metrics ||X||g and ||X||g are equivalent For the purpose of the proof of the main theorem, we fix a riemannian metric g and a basis B = (h1, .., hk ) of harm 1 (M, g). The norm of a harmonic 1-form H will be simply denoted |H | and the norm of a symplectic vector field X will be simply denoted ||X||. 4.2 Proof of theorem 3 We prove first that the set SS ym peo(M,ω)subsetS ym peo(M,ω) is closed under composition and inverse maps. Let hi ∈ SS ym peo(M,ω) i = 1, 2 and let λi be continuous paths in H omeo(M) with λi (0) = id, λi (1) = hi and let Φ n i be ds ym p - Cauchy sequences of symplectic isotopies con- verging C0 to λi . Then Φ n 1 .(Φn 2 )−1 converges C0 to the path λ1(t)(λ2(t)) −1. Here Φn 1 .(Φn 2 )−1(t) = φn 1 (t).(φn 2 (t))−1. 60 Augustin Banyaga CUBO 12, 3 (2010) By definition of the distance ds ym p, Φ n is a ds ym p - Cauchy sequence if and only if both Φ n and (Φn)−1 are D0 - Cauchy and d- Cauchy sequences. Main Lemma: If Φn = (φn t ) and Ψn t = (ψn t ) are ds ym p - Cauchy sequences in I so(M), so is ρn t = φn t ψn t . The proof of the main lemma is very delicate; it will take most of the remaining part of this paper. The estimates are much more involved than in the hamiltonian case, due to the fact that the decomposition of a symplectic isotopy into a hamiltonian one and a harmonic one does not behave nicely with respect to the product of isotopies. It will be enough to prove that ρnt is a D0 - Cauchy sequence. Indeed since (Φ n)−1 and (Ψn)−1 are D0 - Cauchy by assumption, the main lemma applied to their product implies that their product is also D0 Cauchy. Hence (Ψn)−1(Φn)−1 = (ΦnΨn)−1 = (ρn t )−1 is a D0 - Cauchy sequence. This will conclude the proof that SS ym peo(M,ω) is a group. We leave the details to the reader. We will use the following estimate: Proposition 2: There exists a constant E such that for any X ∈ s ym p(M,ω), and H ∈ harm1 (M, g) |H (X )| =: su px∈M |H (x)(X (x))| ≤ E||X||.|H | Proof: Let (h1, .., hr ) be the chosen basis for harmonic 1-forms and let E = maxi E i and E i = su pV (su px∈M |hi (x)(V (x))| where V runs over all symplectic vector fields V such that ||V || = 1. Without loss of generality, we may suppose X 6= 0 and set V = X /||X||. Let H = ∑ λi hi . Then H (X ) = ||X|| ∑ λi hi (V ). Hence |H (X )| ≤ ||X|| ∑ |λi|su px (|hi (x)(V )(x)|) ≤ ||X|| ∑ |λi|E = E||X||.|H |. We will also need the following standard facts: Proposition 3: Let φ be a diffeomorphism, X a vector field and θ a differential form on a smooth manifold M. Then (φ −1 ) ∗ [i X φ ∗ θ] = iφ∗ X θ Proposition 4: If φt,ψt are any isotopies, and if we denote by ρt = φtψt, and by φ t = (φ)−1t then ρ̇t = φ̇t + (φt)∗ψ̇t CUBO 12, 3 (2010) On The Group of Strong Symplectic Homeomorphisms 61 and φ̇ t = −((φ) −1 t )∗(φ̇t) Proposition 5: Let θt be a smooth family of closed 1-forms and φt an isotopy, then φ ∗ t θt −θt = dvt where vt = ˆ t 0 (θt(φ̇s) ◦φs)ds Proof of the Main Lemma: If φt,ψt are symplectic isotopies, and if ρt = φtψt, propositions 3, 4 and 5 give: i(ρ̇t)ω = H Φ t + H Ψ t + dK (Φ,Ψ) (I) where K = K (Φ,Ψ) = uΦt + (u Ψ t ) ◦ (φt) −1 + vt(Φ,Ψ), and vt(Φ,Ψ) = ˆ t 0 (H Ψ t (φ̇s ) ◦φ −1 s )ds. (I I) Let now φn t ,ψn t be Cauchy sequences of symplectic isotopies, and consider the sequence ρnt = φ n t ψ n t . We have: |||ρ̇ n t −ρ̇ m t ||| = ˆ 1 0 |H Φ n t − H Φ m t + H Ψ n t − H Ψ m t |+ osc(K (Φ n ,Ψ n ) − K (Φ m ,Ψ m ))dt ≤ ˆ 1 0 |H Φ n t − H Φ m t )|dt + ˆ 1 0 |H Ψ n t − H Ψ m t )|dt + ˆ 1 0 osc(u Φ n t − u Φ m t )dt + ˆ 1 0 osc(u Ψ n t ) ◦ (φ n t ) −1 − u Ψ m t ◦ (φ m t ) −1 )dt + ˆ 1 0 osc(vt (Φ n ,Ψ n ) − vt(Φ m ,Ψ m )dt = |||φ̇n t −φ̇ m t|||+ ˆ 1 0 |H Ψ n t − H Ψ m t )|dt + A + B where A = ˆ 1 0 osc(u Ψ n t ) ◦ (φ n t ) −1 − u Ψ m t ◦ (φ m t ) −1 )dt and B = ˆ 1 0 osc(vt (Φ n ,Ψ n ) − vt(Φ m ,Ψ m )dt. (I I I) 62 Augustin Banyaga CUBO 12, 3 (2010) We have: A ≤ ˆ 1 0 osc(u Ψ n t ) ◦ (φ n t ) −1 − u Ψ m t ◦ (φ n t ) −1 )dt + ˆ 1 0 osc(u Ψ m t ) ◦ (φ n t ) −1 − (u Ψ m t ) ◦ (φ m t ) −1 )dt = ˆ 1 0 osc(u Ψ n t − u Ψ m t )dt + C where C = ˆ 1 0 osc(u Ψ m t ◦ (φ n t ) −1 − u Ψ m t ◦ (φ m t ) −1 )dt. Hence |||ρ̇ n t −ρ̇ m t ||| ≤ |||φ̇ n t −φ̇ m t ||| + ˆ 1 0 |H Ψ n t − H Ψ m t )|dt + ˆ t 0 osc(u Ψ n t − u Ψ m t )dt + B + C = |||φ̇ n t −φ̇ m t |||+|||ψ̇ n t −ψ̇ m t |||+ B + C We now show that C → 0 when m, n → ∞. Sub-Lemma 1 (Reparametrization Lemma [13]): ∀ǫ ≥ 0,∃m0 such that C = ˆ 1 0 osc(u Ψ m t ◦ (φ n t ) −1 − u Ψ m t ◦ (φ m t ) −1 )dt =: ||u Ψ m t ◦ (φ n t ) −1 − u Ψ m t ◦ (φ m t ) −1 )|| ≤ ǫ if m ≥ m0 and n large enough Remark: This is the “reparametrization lemma” of Oh-Müller [13] (lemma 3.21. (2)). For the convenience of the reader and further references, we include their proof. Proof: For short, we write um for u Ψ m t and µ n t for (φn t )−1. First, there exists m0 large such that ||um −um0 || ≤ ǫ/3 for m ≥ m0, since (um ) is a Cauchy sequence for the distance d(un , um ) = ´ 1 0 osc(un − um )dt. Therefore ||um ◦µ n t − um ◦µ m t ))|| ≤ ||um ◦µ n t − um0 ◦µ n t ))||+||um0 ◦µ n t − um0 ◦µ m t ))||+||um0 ◦µ m t − um ◦µ m t ))|| = ||um − um0 ||+||um0 ◦µ n t − um0 ◦µ m t ))||+||um0 − um|| ≤ (2/3)ǫ+||um0 ◦µ n t − um0 ◦µ m t ))|. By uniform continuity of um0 , there exists a positive δ such that if d(µ m t ,µn t ) ≤ δ, then max osc((um0 ◦µ n t − um0 ◦µ m t )) ≤ ǫ/3. Hence ||um0 ◦µ n t − um0 ◦µ m t ))|| ≤ ǫ/3 for n, m large. Recall that µn t is a d- Cauchy sequence. CUBO 12, 3 (2010) On The Group of Strong Symplectic Homeomorphisms 63 To show that ρ̇n t is a Cauchy sequence, the only thing which is left is to show that B → 0 when n, m → ∞. Let us denote vt(Φ n,Ψn) by vn t , H Ψ n t by H t n or H n and (φ n t )−1 by µn t . For a function on M, we consider the norm |f | = su px∈M |f (x)| We have: |v n t − v m t | = | ˆ t 0 (H t n(µ̇ n s ) ◦µ n s − H t m(µ̇ m s ) ◦µ m s )ds| ≤ ˆ 1 0 |((H t n − H t m)(µ̇ n s )) ◦µ n s |ds + ˆ 1 0 |H t m(µ̇ n s −µ̇ m s )) ◦µ m s |ds + ˆ 1 0 |H t m (µ̇ n s ) ◦µ n s − H t m(µ̇ n s ) ◦µ m s |ds The last integral can be estimated as follows: ˆ 1 0 |H t m(µ̇ n s ) ◦µ n s − H t m (µ̇ n s ) ◦µ m s |ds ≤ ˆ 1 0 |H t m(µ̇ n s ) ◦µ n s − H t m(µ̇ n0 s ) ◦µ n s |ds (1) + ˆ 1 0 |H t m(µ̇ n0 s ) ◦µ n s − H t m(µ̇ n0 s ) ◦µ m s |ds (2) + ˆ 1 0 |H t m(µ̇ n0 s ) ◦µ m s − H t m(µ̇ n s ) ◦µ m s |ds (3) for some integer n0. Proposition 2 gives E|H m|D0((Φ n)−1, (Φn0 )−1) ≤ 2E|H m|D((Φ n), (Φn0 )−1) as an upper bound for (1) and (3). It also gives the following estimates: ˆ 1 0 |((H t n − H t m)(µ̇ n s )) ◦µ n s |ds ≤ E|H t n − H t m| ˆ 1 0 ||µ̇ n s )||ds = E.|H t n − H t m|.l((Φ n ) −1 ) 64 Augustin Banyaga CUBO 12, 3 (2010) and ˆ 1 0 |(H t m(µ̇ n s −µ̇ m s )) ◦µ m s |ds ≤ E.|H t m| ˆ 1 0 ||(µ̇ n s −µ̇ m s )||ds = E|H t m|D0((Φ n ) −1 , (Φ m )) −1 ) ≤ 2E|H t m|D(Φ n ,Φ m ). Therefore, we get the following estimate: |v n t − v m t | ≤ E.|H t n − H t m|l(Φ n ) −1 ) + E|H t m|2(D(Φ n ,Φ m ) + 2D(Φ n ,Φ n0 )) + G where G = ˆ 1 0 |H t m(µ̇ n0 s ) ◦µ n s − H t m(µ̇ n0 s ) ◦µ m s |ds Since osc(vn t − vm t ) ≤ 2|vn t − vm t |, we see that ˆ 1 0 osc(v n t − v m t )dt ≤ 2E(l(Φ n ) −1 ) ˆ 1 0 |H t n − H t m|dt +E2(D(Φ m ,Φ n ) + 2ED(Φ n ,Φ n0 ) ˆ 1 0 |H t m|dt) + ˆ 1 0 Gdt We need the following facts: Sub-Lemma 2 (Reparametrization Lemma): ∀ǫ ≥ 0,∃n0 such that L = ˆ 1 0 Gdt = ˆ 1 0 ( ˆ 1 0 |H t m(µ̇ n0 s ) ◦µ n s − H t m(µ̇ n0 s ) ◦µ m s |ds ) dt ≤ ǫ for n ≥ n0 and m sufficiently large. Proposition 6: l((Φn))−1 and ´ 1 0 |H t m|dt are bounded for every n, m. We finish first the estimate for ´ 1 0 osc(vnt − v m t )dt using sub-lemma 2 and proposition 6. Putting together all the information we gathered, we see that: ˆ 1 0 osc(v n t − v m t )dt ≤ 2E(l(Φ n ) −1 ) ˆ 1 0 |H t n − H t m|dt +E(2D(Φ m ,Φ n )) + 2ED(Φ n ,Φ n0 )( ˆ 1 0 |H t m|dt) + L ≤ 2El((Φ n ) −1 )D(Φ n ,Φ m ) + E(2D(Φ m ,Φ n ) + 2ED(Φ n ,Φ n0 ) ˆ 1 0 |H t m|dt + L Therefore: ˆ 1 0 osc(v n t − v m t )dt → 0 CUBO 12, 3 (2010) On The Group of Strong Symplectic Homeomorphisms 65 when n, m → ∞, and n0 is chosen sufficiently large Now let n0 → ∞ as well.. This finishes the proof of the main lemma. Proof of Sub-Lemma 2: G = ˆ 1 0 |H t m(µ̇ n0 s ) ◦µ n s − H t m(µ̇ n0 s ) ◦µ m s |ds ≤ ˆ 1 0 |H t m(µ̇ n0 s ) ◦µ n s − H t m0 (µ̇ n0 s ) ◦µ n s |ds + ˆ 1 0 |H t m0 (µ̇ n0 s ) ◦µ n s − H t m0 (µ̇ n0 s ) ◦µ m s |ds + ˆ 1 0 |H t m0 (µ̇ n0 s ) ◦µ m s − H t m(µ̇ n0 s ) ◦µ m s |ds for some m0. Exactly like in the proof of sub-lemma 1 G(t, n, m) ≤ 2E|H t m − H t m0 |.(l(Ψ n0 ) −1 ) + F where F = ˆ 1 0 |H t m0 (µ̇ n0 s ) ◦µ n s − H t m0 (µ̇ n0 s ) ◦µ m s |ds By uniform continuity of H tm0 (µ̇ n0 s ), F → 0 when n, m → ∞ since µ n t is Cauchy. By similar arguments as in the sub-lemma 1, G → 0 and hence L → 0 when m, n → ∞ and m0 → ∞. We have just proved that the subset SS ym p(M,ω) of S ymeo(M,ω) is closed under com- position and inversion. This concludes the proof that SS ym peo(M,ω) is a group. The fact that it is arcwise connected in the ambiant topology of H omeo(M) is obvious from the definition. H ameo(M,ω) is a normal subgroup of SS ym peo(M,ω) since it is normal in S ym peo(M,ω) [13]. Let h, g ∈ SS ym peo(M,ω) and let Φn,Ψn be symplectic isotopies which form Cauchy sequences and C0 converge to h, g. By the main lemma the sequence Φn.Ψn.(Φn)−1(Φn)−1 is a Cauchy sequence. It obviously converges C0 to the commutator h gh−1 g−1 ∈ SS ym peo(M,ω). It is a standard fact that Φn.Ψn.(Φn)−1(Ψn)−1 is a hamiltonian isotopy. Indeed let φt and ψt be symplectic isotopies, and let σt = φtψtφ −1 t ψ −1 t , then σ̇t = X t + Yt + Zt +Ut 66 Augustin Banyaga CUBO 12, 3 (2010) with X t = φ̇t , Yt = (φt)∗ψ̇t, Zt = −(φtψtφ −1 t )∗φ̇t, and Ut = −(σt)∗ψ̇t. By proposition 5, i(X t+Zt)ω and i(Yt+Ut)ω are exact 1-forms. Hence σt is a hamiltonnian isotopy. By Proposition 1, the metric D coincides with the one for hamiltonian isotopies. Hence Φ n.Ψn.(Φn)−1(Ψn)−1 is a Cauchy sequence for dham. Therefore: [SS ym peo(M,ω), SS ym peo(M,ω)] ⊂ H ameo(M,ω)]. This concludes the proof of theorem 3 Proof of Theorem 2: We now prove that SS ym peo(M,ω), with the symplectic topology, is a topological group. In fact, we prove that Q ( see section 3) is a topological group. Recall that an element of Q is a couple (γ, V = (H , u) where γ ∈ P H omeo(M), H ∈ L(1,∞)([0, 1], harm1 (M,ω) , u ∈ L(0,1)([0, 1]xM,R), and there ex- ists a ds ym p - Cauchy sequence of symplectic isotopies Φn(t) such that Φn(1) → γ , in the C 0 topology and limn→∞ (H n, un ) = (H , u). Here we wrote H n for H Φn and un for u Φn n . The product and the inverse in Q are given by: (γ, (H , u)).(γ ′ , (H ′ , u ′ )) = (γγ ′ , (H + H ′ , u + u ′ ◦γ+ v)) (γ, (H , u)) −1 = (γ −1 , (−H ,−(u ◦γ+ w)) where v is the limit of the Cauchy sequence vn(t) given by formula (II): vn(t) = ˆ 1 0 (H ′ n(σ̇n(s)) ◦σn (s))ds, with σn(s) = (Φ ′ n(s)) −1. and w the limit of a similar sequence in which σn is replaced by Φn . Part I. Let us first show that the inversion is continuous: let (γk, (H k, uk )) be a sequence converging to (γ, (H , u)), For each k, there is a Cauchy sequence Φkn of symplectic isotopies such that Φkn → γk as n → ∞ in the C 0 topology, H kn → H k, u k n → uk . We need only to show that wk → w, that is (*) l imn,k→∞ ˆ 1 0 |H k n (Φ̇ k n(s)) ◦Φ k n (s) − H n (Φ̇n(s)) ◦Φn (s)|ds = 0. We have the following inequalities : ||Φ̇ k n −Φ̇n|| ≤ ||Φ̇ k n − V k ||+||V k − V ||+||V −Φ̇n|| and each term in the right hand of this inequality → 0 as n, k → ∞. Similarly, |H k n − H n| ≤ |H k n − H k |+|H k − H |+|H − H n| CUBO 12, 3 (2010) On The Group of Strong Symplectic Homeomorphisms 67 and each term in the right hand of this inequality → 0 as n, k → ∞. Formula (*) follows from these inequalities and the techniques developped in this paper (including the reparametrisation lemma). We leave the details to the reader. Part II. Now we prove that the composition is continuous: let (γk, V k = (H k, uk )) and (γ′k, V ′k = (H ′k, u′k )) converging to (γ, (H , u)) and (γ′, (H ′, u′)). By part I, if σ̇kn → U k and σ̇′ k n → U ′k,then by part I, U k → U. Here we denoted by σkn, and σ′ k n respectively (Φ k n) −1, (Φ′kn ) −1. We only need to prove: 1) uk ◦γk → u ◦γ 2) vk → v. The proof of (1) goes along the lines explained in this paper ( including the repara- mareization lemma ) and the details are left tothe reader. The proof of (2) follows from part I and uses the inequalities: ||σ̇ k n −ρ̇n|| ≤ ||σ̇ k n −U k ||+||U k −U||+||U −ρ̇n|| Each of the three parts of the second member of the inequality → 0as n, k → ∞. The details are left tothe reader. This concludes the proof of theorem 2. Appendix: For the convenience of the reader, we give here the proofs of propositons 3, 4, and 5. Proof of Proposition 3: Let θ be a p-form, X a vector field and φ a diffeomorphism. For any x ∈ M and any vector fields Y1, ..Yp−1, we have: (φ−1)∗[i X φ ∗θ](x)(Y1, ..., Yp−1 ) = (i X φ ∗θ)(φ−1(x))(D xφ −1(Y1(x), ...(D xφ −1(Yp−1(x)) = (φ∗θ)(φ−1(x))(Xφ−1(x), D xφ −1(Y1(x)), ...(D xφ −1(Yp−1(x)) = θ(φ(φ−1(x))(Dφ−1(x)φ(Xφ−1(x)), Dφ−1(x)φD xφ −1(Y1(x)), ...Dφ−1 (x)φD xφ −1(Yp−1(x) = θ(x)((φ∗ X )x, Y1(x), ..Yp−1(x)) = (i(φ∗ X )θ)(x)(Y1, .., Yp−1 ) since Dφ−1(x)φD xφ −1 = D x(φφ −1) = id. Therefore (φ−1)∗[i X φ ∗θ] = i(φ∗ X ))θ Proof of Proposition 4: This is just the chain rule. See [10] page 145. 68 Augustin Banyaga CUBO 12, 3 (2010) Proof of proposition 5: For a fixed t, we have d ds φ ∗ s θt = φ ∗ s (Lφ̇s θt), where L X is the Lie derivative in the direction X . Since θ is closed, we have: d ds φ ∗ s θt = φ ∗ s (d iφ̇s θt) = d(φ ∗ s (θt(φ̇s)) = d(θt(φ̇s) ◦φs). Hence for every u φ ∗ uθt −θt = ˆ u 0 d ds φ ∗ s θt ds = d( ˆ u 0 (θt(φ̇s) ◦φs)ds) Now set u = t. Acknowledgement I would like to thank Claudio Cuevas for soliciting this paper for Cubo. 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