Ivancevic_Applied-Diff-Geom

650 Applied Differential Geometry: A Modern Introductionevery fiber of π not just on the the chosen fiber W . This shows firstly thatπ : P → B is symplectic, because there is a well defined symplectic form oneach of its fibers, and secondly that it is Hamiltonian with respect to thisform Ω κ W on the fiber W . Hence H2 (P ) surjects onto H 2 (W ).Now suppose that τ W is any closed connection form on π P : W → F .Because the restriction map H 2 (P ) → H 2 (W ) is surjective, the cohomologyclass [τ W ] is the restriction of a class on P and so, by Thurston’s construction,the form τ W can be extended to a closed connection form τ P for thebundle π P . Therefore the previous argument applies in this case too.Now let us consider the general situation, when π 1 (B) ≠ 0. The proofof the lemma above applies to show that the composite bundle π : P → Bis symplectic with respect to suitable Ω κ W and that it has a symplecticconnection form. However, even though π X : X → B is symplecticallytrivial over the 1-skeleton of B the same may not be true of the compositemap π : P → B. Moreover, in general it is not clear whether triviality withrespect to one form Ω κ W implies it for another. Therefore, we may concludethe following: If(M, ω) → P π P→ X and (F, σ) → X π →XBare Hamiltonian fiber bundles and P is compact, then the composite π =π X ◦ π P : P → B is a symplectic fiber bundle with respect to any form Ω κ Won its fiber W = π −1 (pt), provided that κ is sufficiently large. Moreoverif π is symplectically trivial over the 1−skeleton of B with respect to Ω κ Wthen π is Hamiltonian.In practice, we will apply these results in cases where π 1 (B) = 0. Wewill not specify the precise form on W , assuming that it is Ω κ W for a suitableκ.4.12.2.3 Splitting of Rational CohomologyWe write H ∗ (X), H ∗ (X) for the rational (co)homology of X. Recall that abundle π : P → B with fiber M is said to be c−split ifH ∗ (P ) ∼ = H ∗ (B) ⊗ H ∗ (M).This happens iff H ∗ (M) injects into H ∗ (P ). Dually, it happens iff therestriction map H ∗ (P ) → H ∗ (M) is onto. Note also that a bundle P → Bc−splits iff the E 2 term of its cohomology spectral sequence is a productand all the differentials d k , k ≥ 2, vanish [Lalonde and McDuff (2002)].