Floer Cohomology and Non - Contractible Periodic Orbits

For a closed symplectic manifold (M,ω), a compatible almost complex structure J , a 1-periodic time dependent symplectic vector field Z and a homotopy class of closed curves γ we define a Floer complex based on 1-periodic trajectories of Z in the homotopy class γ. We show how to associate to the above data an invariant, the symplectic torsion, which is an element in the Whitehead group Wh(Λ0), of a Novikov ring Λ0 associated with (M,ω, Z, γ). We prove, that when γ is non-trivial the cohomology of the Floer complex is trivial, but the symplectic torsion can be non-trivial. Using the first fact we prove results about non-contractible 1-periodic trajectories of 1-periodic symplectic vector fields. In this paper we will only prove the statements for closed weakly monotone manifolds, but note that they remain true as formulated for arbitrary closed symplectic manifolds.