  Motivated by the theory of representability classes by submanifolds, we study
the rational homotopy theory of Thom spaces of vector bundles. We first give a
Thom isomorphism at the level of rational homotopy, extending work of
Felix-Oprea-Tanr\'e by removing hypothesis of nilpotency of the base and
orientability of the bundle. Then, we use the theory of weight decompositions
in rational homotopy to give a criterion of representability of classes by
submanifolds, generalising results of Papadima. Along the way, we study issues
of formality and give formulas for Massey products of Thom spaces. Lastly, we
link the theory of weight decompositions with mixed Hodge theory and apply our
results to motivic Thom spaces.
