A Characteristic Number of Hamiltonian Bundles over $S^2$

Details A Characteristic Number of Hamiltonian Bundles over $S^2$ A Characteristic Number of Hamiltonian Bundles over $S^2$

2005-06-09

arxiv.org/abs/math/0506174v2

Mathematics

Symplectic Geometry

Version to appear in Journal of Geometry and Physics

Scientific paper

10.1016/j.geomphys.2005.12.004

Each loop $\psi$ in the group $\text{Ham}(M)$ of Hamiltonian diffeomorphisms of a symplectic manifold $M$ determines a fibration $E$ on $S^2$, whose coupling class \cite{G-L-S} is denoted by $c$. If $VTE$ is the vertical tangent bundle of $E$, we relate the characteristic number $\int_E c_1(VTE)c^n$ with the Maslov index of the linearized flow $\psi_{t*}$ and the Chern class $c_1(TM)$. We give the value of this characteristic number for loops of Hamiltonian symplectomorphisms of Hirzebruch surfaces.

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