Ivancevic_Applied-Diff-Geom

642 Applied Differential Geometry: A Modern IntroductionHam(M)), and set π S = π 1 (Symp 0 ) so that there are fibrations˜Ham → Symp 0F lux→ H 1 (M, R), B(π S ) → B Symp 0 → B Symp 0 .The existence of the first fibration shows that ˜Ham is homotopy equivalentto Symp 0 so that B ˜Ham ≃ B Symp 0 , while the second implies that thereis a fibrationB Symp 0 → B Symp 0 → K(π S , 2),where K(π S , 2) is an Eilenberg–MacLane space. A symplectic bundle overB is equivalent to a homotopy class of maps B → B Symp 0 . If B is2−connected, the composite B → B Symp 0 → K(π S , 2) is null homotopic,so that the map B → B Symp 0 lifts to B Symp 0 and hence to the homotopicspace B ˜Ham. Composing this map B → B ˜Ham with the projectionB ˜Ham → B Ham we get a Hamiltonian structure on the given bundleover B.Equivalently, use the existence of the fibration ˜Ham → Symp 0 →H 1 (M, R) to deduce that the subgroup π 1 (Ham) of ˜Ham injectsinto π 1 (Symp 0 ). This implies that the relative homotopy groupsπ i (Symp 0 , Ham) vanish for i > 1, so thatπ i (B Symp 0 , B Ham) = π i−1 (Symp 0 , Ham) = 0, (i > 2).The desired conclusion now follows by obstruction theory. The second proofdoes not directly use the sequence0 −→ Ham −→ Symp 0 −→ H 1 /Γ ω −→ 0,since the flux group Γ ω may not be a discrete subgroup of H 1 .4.12.1.9 Classification of Hamiltonian StructuresThe previous subsection discussed the question of the existence of Hamiltonianstructures on a given bundle. We now look at the problem of describingand classifying them.Let π : P → B be a symplectic bundle satisfying the above conditionsand fix an identification of (M, ω) with (M b0 , ω b0 ). Let a be anyclosed extension of [ω], γ 1 , . . . , γ k be a set of generators of the first rationalhomology group of B, {c i } the dual basis of H 1 (B) and T 1 , . . . , T k symplectictrivializations round the γ i . Assume for the moment that each classf(T i , a) ∈ H 1 (M b0 ) = H 1 (M) has an extension ˜f(T i , a) to P . Subtracting