Hamiltonian isotopies of relatively exact Lagrangians are orientation-preserving

Abstract

Given a closed, orientable Lagrangian submanifold L in a symplectic manifold (X, ω), we show that if L is relatively exact then any Hamiltonian diffeomorphism preserving L setwise must preserve its orientation. In contrast to previous results in this direction, there are no spin hypotheses on L. Curiously, the proof uses only mod 2 coefficients in its singular and Floer cohomology rings.