Ivancevic_Applied-Diff-Geom

630 Applied Differential Geometry: A Modern Introductionis a trivial bundle and Φ is given by a non–Hamiltonian loop φ inSymp 0 (M), then Φ is in Symp 0 (P, π) but it preserves no Hamiltonianstructure on P sinceΦ ∗ (a) = a + [dt] ⊗ F lux(φ).In general, if we choose a trivialization of P over B 1 , there are exactsequences{Id} → Aut(P, π, {a}) → Symp 0 (P, π) → Hom(π 1 (B), Γ ω ) → {id},{Id} → Ham(P, π, {a}) → Aut(P, π, {a}) → H 1 (M, R)/Γ ω → {0}.In particular, the subgroup of Aut(P, π, {a}) consisting of automorphismsthat belong to Ham(M b0 , ω b0 ) at the base point b 0 retracts toHam(P, π, {a}).4.12.1.3 Marked Hamiltonian StructuresAnother approach to characterizing a Hamiltonian structure is to define itin terms of a structure on the fiber that is preserved by elements of theHamiltonian group [Lalonde and McDuff (2002)].The so–called marked symplectic manifold (M, ω, [L]) is a pair consistingof a closed symplectic manifold (M, ω) together with a marking [L]. HereL is a collection {l 1 , . . . , l k } of loops l i : S 1 → M in M that projectsto a minimal generating set G L = {[l 1 ], . . . , [l k ]} for H 1 (M, Z)/torsion. Amarking [L] is an equivalence class of generating loops L, where L ∼ L ′ iffor each i there is an singular integral 2−chain c i whose boundary modulotorsion is l ′ i − l i such that ∫ c iω = 0.The symplectomorphism group acts on the space L of markings. Moreover,it is easy to check that if a symplectomorphism φ fixes one marking[L] it fixes them all. Hence the groupLHam(M, ω) = LHam(M, ω, [L]) = {φ ∈ Symp(M, ω) : φ ∗ [L] = [L]}independent of the choice of [L]. Its identity component is Ham(M, ω).There is a forgetful map [L] → G L from the space L of markings tothe space of minimal generating sets for the group H 1 (M, Z), and it is nothard to check that its fiber is (R/P) k , where is the image of the periodhomomorphismI [ω] : H 2 (M, Z) → R.