Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 659that the bundles given by restricting P to the different curves countedin n B (pt, pt, c 1 , . . . , c k ; A) are diffeomorphic, since, this being a homotopytheoretic question, we can always replace X by the simply connected spaceX/(X 1 ) in which these curves are homotopic [McDuff (2000)].4.12.2.5 Homotopy Reasons for SplittingIn this section we discuss c−splitting in a homotopy-theoretic context. Recallthat a c−Hamiltonian bundle is a smooth bundle P → B together witha class a ∈ H 2 (P ) whose restriction a M to the fiber M is c−symplectic, i.e.,(a M ) n ≠ 0 where 2n = dim(M). Further a closed manifold M is said to satisfythe hard Lefschetz condition with respect to the class a M ∈ H 2 (M, R)if the following maps are isomorphisms,∪(a M ) k : H n−k (M, R) → H n+k (M, R),(1 ≤ k ≤ n).In this case, elements in H n−k (M) that vanish when cupped with (a M ) k+1are called primitive, and the cohomology of M has an additive basis consistingof elements of the form b ∪ (a M ) l where b is primitive and l ≥ 0. 20Let M → P → B be a c−Hamiltonian bundle such that π 1 (B) actstrivially on H ∗ (M, R). If in addition M satisfies the hard Lefschetz conditionwith respect to the c−symplectic class a M , then the bundle c−splits[Blanchard (1956)].The proof is by contradiction. Consider the Leray spectral sequence incohomology and suppose that d p is the first non zero differential. Then,p ≥ 2 and the E p term in the spectral sequence is isomorphic to the E 2 termand so can be identified with the tensor product H ∗ (B) ⊗ H ∗ (M). Becauseof the product structure on the spectral sequence, one of the differentialsd 0,ip must be nonzero. So there is b ∈ Ep0,i ∼= H i (M) such that d 0,ip (b) ≠ 0.We may assume that b is primitive (since these elements together with a Mgenerate H ∗ (M).) Then b ∪ a n−iMWe can write≠ 0 but b ∪ an−i+1M= 0.d p (b) = ∑ je j ⊗f j , where e j ∈ H ∗ (B) and f j ∈ H l (M) (with l < i).Hence f j ∪a n−i+1M≠ 0 for all j by the Lefschetz property. Moreover, becausethe E p term is a tensor product(d p (b)) ∪ a n−i+1M= ∑ je j ⊗ (f j ∪ a n−i+1M) ≠ 0.20 These manifolds are sometimes called ‘cohomologically Kähler manifolds.’