Reidemeister torsion and integrable Hamiltonian systems (Q1970272)

Let \(N\) be a 4-dimensional smooth symplectic manifold and consider the Hamiltonian system with smooth Hamiltonian \(H\), which in Darboux coordinates has the form \[ {dp_i\over dt}=- {\partial H\over\partial q_i},\;{dq_i \over dt}={\partial H\over\partial P_i}.\tag{1} \] The 3-dimensional level surface \(M=\{H=\) constant\} is called an isoenergy surface and is invariant under the flow defined by the system (1). Recently, the Reidemeister torsion has found interesting applications in dynamical systems theory. Here the authors study the Reidemeister torsion of isoenergy surfaces of an integrable Hamiltonian system (1). The authors use the spectral sequence defined by filtration and following ideas of Witten-Floer they bring into play the orbits connecting the critical submanifolds.