Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 653is the universal bundle. Every compact connected Lie group G is the imageof a homomorphism T × H → G, where the torus T maps onto the identitycomponent of the center of G and H is the semi–simple Lie group correspondingto the commutator subalgebra [Lie(G), Lie(G)] in the Lie algebraLie(G). It is easy to see that this homomorphism induces a surjection onrational homology BT × BH → BG. Therefore, we may suppose thatG = T × H. Let T max = (S 1 ) k be the maximal torus of the semi–simplegroup H. Then the induced map on cohomologyH ∗ (BH) → H ∗ (BT max ) = Q[a 1 , . . . , a k ]takes H ∗ (BH) bijectively onto the set of polynomials in H ∗ (BT max ) thatare invariant under the action of the Weyl group, by the Borel–Hirzebruchtheorem. Hence the mapsBT max → BH and BT × BT max → BGinduce a surjection on homology. Therefore the desired result follows fromthe surjection lemma and the Atiyah–Bott theorem.We have the following lemma [Lalonde and McDuff (2002)]: EveryHamiltonian bundle over a coadjoint orbit c−splits.This is an immediate consequence of the results by [Grossberg andKarshon (1994)] on Bott towers. Recall that a Bott tower is an iteratedfibration of Kähler manifoldsM k → M k−1 → · · · → M 1 = S 2where each map M i+1 → M i is a fibration with fiber S 2 . They show thatany coadjoint orbit X can be blown up to a manifold that is diffeomorphicto a Bott tower M k . Moreover the blow–down map M k → X inducesa surjection on rational homology. Every Hamiltonian bundle over M kc−splits. Hence the result follows from the surjection lemma.Every Hamiltonian bundle over a 3−complex X c−splits. As in theproof of stability given above, we can reduce this to the cases X = S 2 andX = S 3 . The only difference from the stability result is that we now requirethe differentials d 0,q2 , d0,q 3 to vanish for all q rather than just at q = 2.Every Hamiltonian bundle over a product of spheres c−splits, providedthat there are no more than 3 copies of S 1 .By hypothesis,B = ∏ S 2mi × ∏ S 2ni+1 × T k ,i∈Ij∈J