Non-contractible periodic trajectories of symplectic vector fields, Floer cohomology and symplectic torsion

Abstract

For a closed symplectic manifold $(M,\omega)$, a compatible almost complex structure $J$, a 1-periodic time dependent symplectic vector field $Z$ and a homotopy class of closed curves $\gamma$ we define a Floer complex based on 1-periodic trajectories of $Z$ in the homotopy class $\gamma$. We suppose that the closed 1-form $i_{Z_t}\omega$ represents a cohomology class $\beta(Z):=\beta$, independent of $t$. We show how to associate to $(M,\omega,\gamma,\beta)$ and to two pairs $(Z_i,J_i)$, $i=1,2$ with $\beta(Z_i)=\beta$ an invariant, the relative symplectic torsion, which is an element in the Whitehead group $Wh(\Lambda_0)$, of a Novikov ring $\Lambda_0$ associated with $(M,\omega,Z,\gamma)$. If the cohomology of the Floer complex vanishes or if $\gamma$ is trivial we derive an invariant, the symplectic torsion for any pair $(Z,J)$. We prove, that when $\beta(\gamma)\neq 0$, or when $\gamma$ is non-trivial and $\beta$ is 'small', the cohomology of the Floer complex is trivial, but the symplectic torsion can be non-trivial. Using the first fact we conclude results about non-contractible 1-periodic trajectories of 1-periodic symplectic vector fields. In this version we will only prove the statements for closed weakly monotone manifolds, but note that they remain true as formulated for arbitrary closed symplectic manifolds.