Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 633question [Lalonde and McDuff (2002)]: Does any Hamiltonian fiber bundleover a compact CW–complex c−split?A special case is when the structural group of P → B can be reducedto a compact Lie subgroup G of Ham(M). Here c−splitting over any basefollows from the work of Atiyah–Bott [Atiyah and Bott (1984)]. In thiscontext, one usually discusses the universal Hamiltonian G−bundle withfiber MM −→ M G = EG × G M −→ BG.The cohomology of P = M G is known as the equivariant cohomologyHG ∗ (M) of M. Atiyah–Bott show that if G is a torus T that acts in aHamiltonian way on M then the bundle M T → BT is c−split. The resultfor a general compact Lie group G follows by standard arguments [Lalondeand McDuff (2002)].The following theorem describes conditions on the base B that implyc−splitting. Let (M, ω) be a closed symplectic manifold, and M ↩→ P → Ba bundle with structure group Ham(M) and with base a compact CW–complex B. Then the rational cohomology of P splits in each of the followingcases:(i) the base has the homotopy type of a coadjoint orbit or of a product ofspheres with at most three of dimension 1;(ii) the base has the homotopy type of a complex blow up of a product ofcomplex projective spaces;(iii) dim(B) ≤ 3.Case (ii) is a generalization of the foundational example B = S 2 and isproved by similar analytic methods. The idea is to show that the mapι : H ∗ (M) → H ∗ (P ) is injective by showing that the image ι(a) in Pof any class a ∈ H ∗ (M) can be detected by a nonzero Gromov–Witteninvariant of the form n P (ι(a), c 1 , . . . , c n ; σ), where c i ∈ H ∗ (P ) and σ ∈H 2 (P ) is a spherical class with nonzero image in H 2 (B). The proof shouldgeneralize to the case when all one assumes about the base is that there is anonzero invariant of the form n B (pt, pt, c 1 , . . . , c k ; A) [Lalonde et al. (1998);Lalonde et al. (1999)].The proofs of parts (i) and (iii) start from the fact of c−splitting overS 2 and proceed using purely topological methods. The following fact aboutcompositions of Hamiltonian bundles is especially useful. Let M ↩→ P →B be a Hamiltonian bundle over a simply connected base B and assume