  We show that the transfer map on Floer homotopy types associated to an exact
Lagrangian embedding is an equivalence. This provides an obstruction to
representing isotopy classes of Lagrangian immersions by Lagrangian embeddings,
which, unlike previous obstructions, is sensitive to information that cannot be
detected by Floer cochains. We show this by providing a concrete computation in
the case of spheres.
