Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 635spheres are c−split [Lalonde and McDuff (2002)].In particular, looking at the action on H 0 (M), we see that the pointevaluation mapev : Ham(M) → M :ψ ↦→ ψ(x)induces the trivial map on rational (co)homology. It also induces the trivialmap on π 1 . However, the map on π k , k > 1, need not be trivial. To see this,consider the action of Ham(M) on the symplectic frame bundle SF r(M)of M and the corresponding point evaluation maps. The obvious action ofSO(3) ≃ Ham(S 2 ) on SF r(S 2 ) ≃ RP 3 induces an isomorphismH 3 (SO(3)) ∼ = H 3 (SF r(S 2 )),showing that these evaluation maps are not homologically trivial. Moreover,its composite with the projection SF r(S 2 ) → S 2 gives rise to a nonzeromapπ 3 (SO(3)) = π 3 (Ham(S 2 )) → π 3 (S 2 ).Thus the corresponding Hamiltonian fibration over S 4 with fiber S 2 , thoughc−split, does not have a section.Note, however, that the extended evaluation mapπ 2l (X X ) ev → π 2l (X) → H 2l (X, Q), l > 0,is always zero, if X is a finite CW complex and X X is its space of self–maps. Indeed, because the cohomology ring H ∗ (X X , Q) is freely generatedby elements dual to π ∗ (X X ) ⊗ Q, there would otherwise be an elementa ∈ H 2l (X) that would pull back to an element of infinite order in thecohomology ring of the H−space X X . Hence a itself would have to haveinfinite order, which is impossible. A more delicate argument shows thatthe integral evaluation π 2l (X X ) → H 2l (X, Z) is zero [Gottlieb (1975)].A Hamiltonian automorphism of the product Hamiltonian bundle B ×M → B is simply a map B → B × Ham(M) of the form b ↦→ (b, φ b ). IfB is a closed manifold we will see that any Hamiltonian automorphism ofthe product bundle acts as the identity map on the rational cohomologyof B × M. The natural generalization of this result would claim that aHamiltonian automorphism of a bundle P acts as the identity map on therational cohomology of P . We do not know yet whether this is true ingeneral. However, we can show that it is closely related to the c−splitting