Ivancevic_Applied-Diff-Geom

638 Applied Differential Geometry: A Modern Introductioncalculation shows that the Φ s are symplectomorphisms. Given a 1−cycleδ : S 1 → M in the fiber M over 1 ∈ ∂D 2 , consider the closed 2−cycle thatis the union of the following two cylinders:c 1 : [0, 1] × S 1 → ∂D 2 × M : (s, t) ↦→ (e 2πis , Φ s (δ(t))),c 2 : [0, 1] × S 1 → 1 × M : (s, t) ↦→ (1, Φ 1−s (δ(t))).This cycle is obviously contractible. Hence,τ(c 1 ) = −τ(c 2 ) = F lux({Φ s })(δ).But τ(c 1 ) = 0 since the characteristics of τ| ∂P are tangent to c 1 . Applyingthis to all δ, we see that the holonomy round ∂D 2 has zero flux and so isHamiltonian.If π 1 (B) = 0 then a symplectic bundle π : P → B is Hamiltonian iff theclass [ω b ] ∈ H 2 (M) extends to a ∈ H ∗ (P ).Suppose first that the class a exists. We can work in the smooth category.Then Thurston’s convexity argument allows us to construct a closedconnection form τ on P and hence a horizontal distribution Hor τ . Theprevious lemma shows that the holonomy around every contractible loopin B is Hamiltonian. Since B is simply connected, the holonomy round allloops is Hamiltonian. Using this, it is easy to reduce the structural group ofP → B to Ham(M). For more details, see [McDuff and Salamon (1998)].Next, suppose that the bundle is Hamiltonian. We need to show thatthe fiber symplectic class extends to P . The proof in [McDuff and Salamon(1998)] does this by the method of [Guillemin et. al. (1998)] and constructsa closed connection form τ, called the coupling form, starting from a connectionwith Hamiltonian holonomy. This construction uses the curvatureof the connection and is quite analytic. In contrast, we shall now use topologicalarguments to reduce to the cases B = S 2 and B = S 3 . These casesare then dealt with by elementary arguments.Consider the Leray–Serre cohomology spectral sequence for M → P →B. Its E 2 term is a product: E p,q2 = H p (B) ⊗ H q (M). 18 We need to showthat the class [ω] ∈ E 0,22 survives into the E ∞ term, which happens iff it isin the kernel of the two differentials d 0,22 , d0,2 3 . Nowd 0,22 : H 2 (M) → H 2 (B) ⊗ H 1 (M)is essentially the same as the flux homomorphism. More precisely, if c :S 2 → B represents some element (also called c) in H 2 (B), then the pullback18 Here H ∗ denotes cohomology over R.