Ivancevic_Applied-Diff-Geom

660 Applied Differential Geometry: A Modern IntroductionBut this is impossible since this element is the image via d p of the trivialelement b ∪ a n−i+1M[Lalonde and McDuff (2002)].Here is a related argument due to [Kedra (2000)]. Every Hamiltonianbundle with 4-dimensional fiber c−splits.Consider the spectral sequence as above. We know that d 2 = 0 andd 3 = 0. Consider d 4 . We just have to check that d 0,34 = 0 since d 0,i4 = 0for i = 1, 2 for dimensional reasons, and = 0 for i = 4 since the top classsurvives.Suppose d 4 (b) ≠ 0 for some b ∈ H 3 (M). Let c ∈ H 1 (M) be suchthat b ∪ c ≠ 0. Then d 4 (c) = 0 and d 4 (b ∪ c) = d 4 (b) ∪ c ≠ 0 sinced 4 (b) ∈ H 4 (B) ⊗ H 0 (M). But we need d 4 (b ∪ c) = 0 since the top classsurvives. So d 4 = 0 and then d k = 0, k > 4 for reasons of dimension.Here is an example of a c−Hamiltonian bundle over S 2 that is notc−split. This shows that c−splitting is a geometric rather than a topological(or homotopy–theoretic) property.Observe that if S 1 acts on manifolds X, Y with fixed points p X , p Ythen we can extend the S 1 action to the connected sum X#Y opp at p X , p Ywhenever the S 1 actions on the tangent spaces at p X and p Y are the same. 21Now let S 1 act on X = S 2 × S 2 × S 2 by the diagonal action in the firsttwo spheres (and trivially on the third) and let the S 1 action on Y be theexample constructed in [McDuff (1988)] of a non–Hamiltonian S 1 actionthat has fixed points. The fixed points in Y form a disjoint union of 2−toriand the S 1 action in the normal directions has index ±1. In other words,there is a fixed point p Y in Y at which we can identify T pY Y with C ⊕C ⊕ C, where θ ∈ S 1 acts by multiplication by e iθ in the first factor, bymultiplication by e −iθ in the second and trivially in the third. Since thereis a fixed point on X with the same local structure, the connected sumZ = X#Y opp does support an S 1 − action. Moreover Z is a c−symplecticmanifold. There are many possible choices of c−symplectic class: underthe obvious identification of H 2 (Z) with H 2 (X) + H 2 (Y ) we will take thec−symplectic class on Z to be given by the class of the symplectic form onX.Let P X → S 2 , P Y → S 2 and P Z → S 2 be the corresponding bundles.Then P Z can be thought of as the connect sum of P X with P Y along thesections corresponding to the fixed points. By analyzing the correspondingMayer–Vietoris sequence, it is easy to check that the c−symplectic class onZ extends to P Z . Further, the fact that the symplectic class in Y does not21 Here Y opp denotes Y with the opposite orientation.