Ivancevic_Applied-Diff-Geom

626 Applied Differential Geometry: A Modern Introductionplectic. A simple argument due to [McDuff and Salamon (1998)] showsthat the cohomological condition (ii) above is equivalent to the existenceof a closed extension τ of the forms ω b . Condition (i) is then equivalent torequiring that the holonomy of ▽ τ around any loop in B belongs to theidentity component Symp 0 (M) of Symp(M). Hence the above result canbe rephrased in terms of such closed extensions τ as follows: A symplecticbundle π : P → B is Hamiltonian iff the forms ω b on the fibers have aclosed extension τ such that the holonomy of ▽ τ around any loop in B liesin the identity component Symp 0 (M) of Symp(M).This is a slight extension of a result of [Guillemin et. al. (1998)], whocalled a specific choice of τ the coupling form. As we show below, theexistence of τ is the key to the good behavior of Hamiltonian bundles undercomposition.When M is simply connected, Ham(M) is the identity componentSymp 0 (M) of Symp(M), and so a symplectic bundle is Hamiltonian iffcondition (i) above is satisfied, i.e., iff it is trivial over the 1−skeleton B 1 .In this case, as observed by [Gotay et. al. (1983)], it is known that (i)implies (ii) for general topological reasons to do with the behavior of evaluationmaps. More generally, (i) implies (ii) for all symplectic bundles withfiber (M, ω) iff the flux group Γ ω vanishes.4.12.1.2 Hamiltonian StructuresThe question then arises as to what a Hamiltonian structure on a fiberbundle actually is [Lalonde et al. (1998); Lalonde et al. (1999); Lalondeand McDuff (2002)]. That is, how many Hamiltonian structures can oneput on a given symplectic bundle π : P → B? And, what does one meanby an automorphism of such a structure?In homotopy theoretic terms, a Hamiltonian structure on a symplecticbundle π : P → B is simply a lift ˜g to B Ham(M) of the classifying mapg : B → B Symp(M, ω) of the underlying symplectic bundle, i.e., it is ahomotopy commutative diagram