On The Group of Strong Symplectic Homeomorphisms

AUGUSTIN BANYAGA
2010 Cubo (Temuco)  
We generalize the "hamiltonian topology" on hamiltonian isotopies to an intrinsic "symplectic 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 contains Hameo(M,ω) as a normal subgroup and coincides with it when
more » ... cides with it when M is simply connected. 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 "topología 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 coincide 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). Augustin Banyaga CUBO 12, 3 (2010) Key words and phrases: Hamiltonian homeomorphisms, hamiltonian topology, symplectic topology, stromg symplectic homeomorphisms, C 0 symplectic topology. Math. Subj. Class.: MSC2000:53D05; 53D35. Introduction No natural metric on the group S ymp(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 ymp(M,ω) ( theorem 1'). We use this natural topology on S ymp(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 ymp(M,ω) of symplectomorphisms of a closed symplectic manifold (M,ω) is C 0 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 C 0 symplectic topology underlying the symplectic geometry of a closed symplectic manifold (M,ω). According to Oh-Müller ([13]), the automorphism group of the C 0 symplectic topology is the closure of the group S ymp(M,ω) in the group Homeo(M) of homeomorphisms of M endowed with the C 0 topology. That group, denoted S ympeo(M,ω) has been called the group of symplectic homeomorphisms: S ympeo(M,ω) =: S ymp(M,ω). The C 0 topology on Homeo(M) coincides with the metric topology coming from the metric d(g, h) = max(su p x∈M d 0 (g(x), h(x)), su p x∈M d 0 (g −1 (x), h −1 (x)) where d 0 is a distance on M induced by some riemannian metric [3]. On the space PHomeo(M) of continuous paths γ : [0,1] → Homeo(M), one has the distance d(γ,µ) = su p t∈[0,1] d(γ(t),µ(t)). Consider the space PHam(M) of all isotopies 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. and Φ 0 H = id. Recall that X H is uniquely defined by the equation The set of time one maps of all hamiltonian isotopies {Φ t H } form a group, denoted Ham(M,ω) and called the group of hamiltonian diffeomorphisms. Definition: The hamiltonian topology [13] on PHam(M) is the metric topology defined by the distance and the oscillation of a function u is osc(u) = max x∈M u(x) − min x∈M u(x). Let Hameo(M,ω) denote the space of all homeomorphisms h such that there exists a continuous path λ ∈ PHomeo(M) such that λ(0) = id, λ(1) = h and there exists a Cauchy sequence (for the d ham distance) of hamiltonian isotopies Φ H n , which C 0 converges to λ ( in the d metric). The following is the first important theorem in the C 0 symplectic topology [13]: Theorem (Oh-Müller): The set Hameo(M,ω) is a topological group. It is a normal subgroup of the identity component S ympeo 0 (M,ω) in S ympeo(M,ω). If H 1 (M,R) = 0, then Hameo(M,ω) is strictly contained in S ympeo 0 (M,ω). Remark: It is still unknown in general if the inclusion Hameo(M,ω) ⊂ S ympeo 0 (M,ω) is strict. The group Hameo(M,ω) is the topological analogue of the group Ham(M,ω) of hamiltonian diffeomorphisms.
doi:10.4067/s0719-06462010000300004 fatcat:wln6e5entjfxtk2ogvcu2wfemm