Reidemeister torsion and integrable Hamiltonian systems (Q2761487)

A round Morse function \(f: M \to \mathbb{R}\) is one whose critical points form nondegenerate circles. The function \(f\) is a Bott integral on a symplectic \(4\)-manifold \(M\) if there is a Hamiltonian function \(H: M \to \mathbb{R}\) and \(f\) Poisson commutes with \(H\). The existence of such an \(f\) implies that the Hamiltonian system is integrable. With these hypotheses, the authors create a Floer-like complex based on connecting orbits which are gradient lines of \(f\) connecting critical manifolds. They then define a Reidemeister torsion of the compact energy levels of the system and relate it to the critical circles and connecting orbits associated to \(f\). (Note for the uninitiated: there is a misprint in Hamilton's equations on page 1.).