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We prove that (under appropriate orientation conditions, depending on $R$) aHamiltonian isotopy $\psi^1$ of a symplectic manifold $(M, \omega)$ fixing arelatively exact Lagrangian $L$ setwise must act trivially on $R_*(L)$, where$R_*$ is some generalised homology theory. We use a strategy inspired by thatof Hu, Lalonde and Leclercq (\cite{Hu-Lalonde-Leclercq}), who proved ananalogous result over $\mathbb{Z}/2$ and over $\mathbb{Z}$ under strongerorientation assumptions. However the differences in our approaches let usdeduce that if $L$ is a homotopy sphere, $\psi^1|_L$ is homotopic to theidentity. Our technical set-up differs from both theirs and that of Cohen,Jones and Segal (\cite{Cohen-Jones-Segal, Cohen}). We also prove (under similarconditions) that $\psi^1|_L$ acts trivially on $R_*(\mathcal{L} L)$, where$\mathcal{L} L$ is the free loop space of $L$. From this we deduce that when$L$ is a surface or a $K(\pi, 1)$, $\psi^1|_L$ is homotopic to the identity.Using methods of \cite{Lalonde-McDuff}, we also show that given a family ofLagrangians all of which are Hamiltonian isotopic to $L$ over a sphere or atorus, the associated fibre bundle cohomologically splits over $\mathbb{Z}/2$.

Families of relatively exact Lagrangians, free loop spaces and generalised homology