C^0 Lagrangian Monodromy

We prove that (under appropriate orientation assumptions), the action of a Hamiltonian homeomorphism φon the cohomology of a relatively exact Lagrangian fixed by φis the identity. This extends results of Hu-Lalonde-Leclercq and the author in the setting of Hamiltonian diffeomorphisms. We also prove a similar result regarding the action of φon relative cohomology.