Ivancevic_Applied-Diff-Geom

644 Applied Differential Geometry: A Modern Introductionlifts are in bijective correspondence with the elements of π 0 (Symp) and socorrespond to an identification (up to symplectic isotopy) of (M, ω) withthe fiber (M b0 , ω b0 ) at the base point b 0 (recall that B is always assumedto be connected).To understand the full lift ˜g, recall the above exact sequence{Id} −→ Ham(M, ω) −→ Symp 0 (M, ω) F lux−→ H 1 (M, R)/Γ ω −→ {0}.If Γ ω is discrete, then the space H 1 (M, R)/Γ ω is homotopy equivalent to atorus and we can investigate the liftings ˜g by homotopy theoretic argumentsabout the fibrationH 1 (M, R)/Γ ω → B Ham(M, ω) → B Symp 0 (M, ω).Now, suppose that a symplectic bundle π : P → B is given that satisfiesthe above conditions. Fix an identification of (M, ω) with (M b0 , ω b0 ).We have to show that lifts from B Symp 0 to B Ham are in bijective correspondencewith equivalence classes of normalized extensions a of the fibersymplectic class. There is a lift iff there is a normalized extension classa. Therefore, it remains to show that the equivalence relations correspond.The essential reason why this is true is that the induced mapπ i (Ham(M, ω)) → π i (Symp 0 (M, ω))is an injection for i = 1 and an isomorphism for i > 1. This, in turn, followsfrom the exactness of the sequence (∗).{Id} −→ Ham(M, ω) −→ Symp 0 (M, ω) F lux−→ H 1 (M, R)/Γ ω −→ {0}.Let us spell out a few more details, first when B is simply connected.Then the classifying map from the 2−skeleton B 2 to B Symp 0 has a lift toB Ham iff the image of the induced mapπ 2 (B 2 ) → π 2 (B Symp 0 (M)) = π 1 Symp 0 (M)lies in the kernel of the flux homomorphismF lux : π 1 (Symp 0 (M)) −→ Γ ω .Since π 1 (Ham(M, ω)) injects into π 1 (Symp 0 (M, ω)), there is only one suchlift up to homotopy. Standard arguments now show that this lift can beextended uniquely to the rest of B. Hence in this case there is a uniquelift. Correspondingly there is a unique equivalence class of extensions a[Lalonde and McDuff (2002)].