Ivancevic_Applied-Diff-Geom

640 Applied Differential Geometry: A Modern IntroductionThe first observation is that this integral is a constant c that is independentof x, since the maps Ψ x : S 2 → M are all homotopic. Secondly, recall thatfor any Hamiltonian vector fields X, Y on M∫∫ω(X, Y )ω n = n ω(X, ·)ω(Y, ·)ω n−1 = 0,MMsince ω(X, ·), ω(Y, ·) are exact 1−forms. Taking X s,t = X s,t (Ψ x (s, t)) andsimilarly for Y , we have∫(∫)c ω n = ω(X s,t , Y s,t ) ω∫I n ds dt = 0.2MHence c = 0. This lemma can also be proved by purely topological methods[Lalonde and McDuff (2002)].Suppose that π B : P → B is Hamiltonian. It is classified by a mapB → B Ham(M). Because B Ham(M) is simply connected this factorsthrough a map C → B Ham(M), where C is obtained by collapsing the1−skeleton of B to a point. In particular condition (i) is satisfied. Toverify (ii), let π C : Q → C be the corresponding Hamiltonian bundle, sothat there is a commutative diagramP✲ Qπ Bπ C❄❄B ✲ C = B/B 1There is a class a C ∈ H 2 (Q) that restricts to [ω] on the fibers. Its pullbackto P is the desired class a.Conversely, suppose that conditions (i) and (ii) are satisfied. By (i),the classifying map B → B Symp(M) factors through a map f : C →B Symp(M), where C is as above. This map f depends on the choice of asymplectic trivialization of π over the 1−skeleton B 1 of B. We now showthat f can be chosen so that (ii) holds for the associated symplectic bundleQ f → C.We need to show that the differentials (d C ) 0,22 , (d C) 0,23 in the spectralsequence for Q f → C both vanish on [ω]. Let· · · → C k (B) ∂ → C k−1 (B) → · · ·be the cellular chain complex for B, and choose 2−cells e 1 , . . . , e k in Bwhose attaching maps α 1 , . . . , α k form a basis over Q for the image of ∂