Ivancevic_Applied-Diff-Geom

662 Applied Differential Geometry: A Modern IntroductionAn element Φ ∈ Ham(P, π) is a map of the formΦ : B × M → B × M :(b, x) ↦→ (b, Φ b (x)),where Φ b ∈ Ham(M) for all b ∈ B. Let us denote the induced mapB ×M → M : (b, x) ↦→ Φ b (x) by α Φ . The previous proposition implies thatif B is a closed manifold of dimension > 0, or, more generally, if it carriesa fundamental cycle [B] of degree > 0,(α Φ ) ∗ ([B] ⊗ m) = Tr [B] (m) = 0, for all m ∈ H ∗ (M).We can also think of Φ : B × M → B × M as the compositeB × M diag B×Id M−→IdB × B × MB ×α−→ΦB × M.The diagonal class in B × B can be written as[B] ⊗ [pt] + ∑ i∈Ib i ⊗ b ′ i, where b i , b ′ i ∈ H ∗ (B) with dim(b ′ i) > 0.HenceΦ ∗ ([B] ⊗ m) = [B] ⊗ m + ∑ i∈Ib i ⊗ Tr b ′i(m) = [B] ⊗ m.More generally, given any class b ∈ H ∗ (B), represent it by the image of thefundamental class [X] of some polyhedron under a suitable map X → Band consider the pullback bundle P X → X. Since the class Φ ∗ ([X] ⊗ m)is represented by a cycle in X × M for any m ∈ H ∗ (M), we can work outwhat it is by looking at the pullback of Φ to X × M. The argument abovethen applies to show that Φ ∗ ([X] ⊗ m) = [X] ⊗ m whenever b has degree> 0. Thus Φ ∗ = Id on all cycles in H ∗>0 (B) ⊗ H ∗ (M). However, it clearlyacts as the identity on H 0 (B) ⊗ H ∗ (M) since the restriction of Φ to anyfiber is isotopic to the identity.We now show that there is a close relation between this question andthe problem of c−splitting of bundles. Given an automorphism Φ of asymplectic bundle M → P → B we define P Φ = (P × [0, 1])/Φ to bethe corresponding bundle over B × S 1 . If the original bundle and theautomorphism are Hamiltonian, so is P Φ → B × S 1 , though the associatedbundle P Φ → B × S 1 → S 1 over S 1 will not be, except in the trivial casewhen Φ is in the identity component of Ham(P, π).Assume that a given Hamiltonian bundle M → P → B c−splits. Thena Hamiltonian automorphism Φ ∈ Ham(P, π) acts trivially (i.e., as the