Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 647for t < 1 but is non–Hamiltonian at t = 1.We have the following result [Lalonde and McDuff (2002)]: A Hamiltonianbundle π : P → B is stable iff the restriction map H 2 (P ) → H 2 (M)is surjective. If π : P → B is Hamiltonian with respect to ω ′ then [ω ′ ] is inthe image of H 2 (P ) → H 2 (M). If π is stable, then [ω ′ ] fills out a neighborhoodof [ω] which implies surjectivity. Conversely, suppose that we havesurjectivity. To check (i) let γ : S 1 → B be a loop in B and suppose thatγ ∗ (P ) is identified symplectically with the product bundle S 1 × (M, ω).Let ω t , 0 ≤ t ≤ ε, be a (short) smooth path with ω 0 = ω. Then, becauseP → B has the structure of an ω t −symplectic bundle for each t, each fiberM b has a corresponding smooth family of symplectic forms ω b,t of the formg ∗ b,t ψ∗ b(ω t ), where ψ b is a symplectomorphism (M b , ω b ) → (M, ω). Hence,for each t, γ ∗ (P ) can be symplectically identified with∪ s∈[0,1] ({s} × (M, g ∗ s,t(ω t ))), where g ∗ 1,t(ω t ) = ω tand the g s,t lie in an arbitrarily small neighborhood U of the identity inDiff(M). By Moser’s homotopy argument, we can choose U so small thateach g 1,t is isotopic to the identity in the group Symp(M, ω t ).Now, the pullback of a stable Hamiltonian bundle is stable [Lalonde andMcDuff (2002)]. Suppose that P −→ B is the pullback of P ′ −→ B ′ via B−→ B ′ so that there is a commutative diagramP ✲ P ′❄B ✲ ❄B ′By hypothesis, the restriction H 2 (P ′ ) −→ H 2 (M) is surjective. But thismap factors as H 2 (P ′ ) −→ H 2 (P ) −→ H 2 (M). Hence H 2 (P ) −→ H 2 (M) isalso surjective. Fe have the following lemma:(i) Every Hamiltonian bundle over S 2 is stable.(ii) Every symplectic bundle over a 2−connected base B is Hamiltonianstable.(i) holds because every Hamiltonian bundle over S 2 is c−split, in particularthe restriction map H 2 (P ) → H 2 (M) is surjective .The above theorem states that every Hamiltonian bundle is stable. Toprove this, first observe that we can restrict to the case when B is simplyconnected. For the map B → B Ham(M) classifying P factors through amap C → B Ham(M), where C = B/B 1 as before, and the stability of the