Hamiltonian isotopies of relatively exact Lagrangians are orientation-preserving

TL;DR

This paper proves that Hamiltonian isotopies of relatively exact Lagrangians preserve orientation without requiring spin conditions, using mod-2 coefficients in cohomology theories.

Contribution

It establishes orientation preservation for Hamiltonian isotopies of relatively exact Lagrangians without spin assumptions, utilizing mod-2 cohomology methods.

Findings

Hamiltonian diffeomorphisms setwise preserving a relatively exact Lagrangian preserve orientation.

No spin hypotheses are needed for the main orientation-preservation result.

The proof employs mod-2 coefficients in singular and Floer cohomology rings.

Abstract

Given a closed, orientable Lagrangian submanifold $L$ in a symplectic manifold $(X, \omega)$, 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.