Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 651The following surjection lemma is obvious but useful: Consider a commutativediagramP ′✲ PB ′❄❄✲ Bwhere P ′ is the induced bundle. Then:(i) If P → B is c−split so is P ′ → B ′ .(ii) If P ′ → B ′ is c−split and H ∗ (B ′ ) → H ∗ (B) is surjective, thenP → B is c−split.To prove (i), we can use the fact that P → B is c−split iff the mapH ∗ (M) → H ∗ (P ) is injective. To prove (ii), the induced map on theE 2 −term of the cohomology spectral sequences is injective. Therefore theexistence of a nonzero differential in the spectral sequence P → B impliesone for the pullback bundle P ′ → B ′ .As a corollary, suppose that P → W is a Hamiltonian fiber bundle overa symplectic manifold W and that its pullback to some blowup Ŵ of W isc−split. Then P → W is c−split.This follows immediately from (ii) above since the map H ∗ (Ŵ ) →H ∗ (W ) is surjective.If (M, ω) → π P → B is a compact Hamiltonian bundle over a simplyconnected CW–complex B and if every Hamiltonian fiber bundle over Mand B is c−split, then every Hamiltonian bundle over P is c−split.Let π E : E → P be a Hamiltonian bundle with fiber F and letF → W → M be its restriction over M. Then by assumption the latterbundle c−splits so that H ∗ (F ) injects into H ∗ (W ). The above compositionlemma implies that the composite bundle E → B is Hamiltonian with fiberW and therefore also c−splits. Hence H ∗ (W ) injects into H ∗ (E). ThusH ∗ (F ) injects into H ∗ (E), as required.If Σ is a closed orientable surface then any Hamiltonian bundle overS 2 × . . . × S 2 × Σ is c−split.Consider any degree one map f from Σ → S 2 . Because Ham(M, ω)is connected, B Ham(M, ω) is simply connected, and therefore any homotopyclass of maps from Σ → B Ham(M, ω) factors through f. Thus anyHamiltonian bundle over Σ is the pullback by f of a Hamiltonian bundleover S 2 . Because such bundles c−split over S 2 , the same is true over Σ.Any Hamiltonian bundle over S 2 × . . . × S 2 × S 1 is c−split. For each