Ivancevic_Applied-Diff-Geom

654 Applied Differential Geometry: A Modern Introductionwhere n i > 0 and 0 ≤ k ≤ 3. SetB ′ = ∏ i∈ICP mi × ∏ j∈JCP ni × T |J| × T l ,where l = k if k +|J| is even, and = k +1 otherwise. Since CP ni ×S 1 mapsonto S 2ni+1 by a map of degree 1, there is a homology surjection B ′ → Bthat maps the factor T l to T k . By the surjection lemma, it suffices to showthat the pullback bundle P ′ → B ′ is c−split.Consider the fibrationT |J| × T l → B ′ → ∏ i∈ICP mi × ∏ j∈JCP ni .Since |J|+l is even, we can think of this as a Hamiltonian bundle. Moreover,by construction, the restriction of the bundle P ′ → B ′ to T |J| × T l is thepullback of a bundle over T k , since the map T |J| → B is nullhomotopic.(Note that each S 1 factor in T |J| goes into a different sphere in B.) Becausek ≤ 3, the bundle over T k c−splits. Hence we can conclude that P ′ → B ′c−splits.Every Hamiltonian bundle whose fiber has cohomology generated by H 2is c−split. This is an immediate consequence of the stability theorem.Any Hamiltonian fibration c−splits if its base B is the image of a homologysurjection from a product of spheres and projective spaces, providedthat there are no more than three S 1 factors. One can also consider iteratedfibrations of projective spaces, rather than simply products. However, wehave not yet managed to deal with arbitrary products of spheres. In orderto do this, it would suffice to show that every Hamiltonian bundle over atorus T m c−splits. This question has not yet been resolved for m ≥ 4.4.12.2.4 Hamiltonian Bundles and Gromov–Witten InvariantsWe begin by sketching an alternate proof that every Hamiltonian bundleover B = CP n is c−split that generalizes the arguments in [McDuff (2000)].We will use the language of [McDuff (1999)], which is based on the Liu–Tian [Liu and Tian (1998)] approach to general Gromov–Witten invariants.Clearly, any treatment of general Gromov–Witten invariants could be usedinstead.We have the fundamental result: [Lalonde and McDuff (2002)] EveryHamiltonian bundle over CP n is c−split. To prove this, the basic idea is toshow that the inclusion ι : H ∗ (M) → H ∗ (P ) is injective by showing that for