  For each positive integer $n$, we give a geometric description of the stably
trivial elements of the group $\pi_n U_n/O_n$. In particular, we show that all
such elements admit representatives whose tangencies with respect to a fixed
Lagrangian plane consist only of folds. By the h-principle for the
simplification of caustics, this has the following consequence: if a Lagrangian
distribution is stably trivial from the viewpoint of a Lagrangian homotopy
sphere, then by an ambient Hamiltonian isotopy one may deform the Lagrangian
homotopy sphere so that its tangencies with respect to the Lagrangian
distribution are only of fold type. Thus the stable triviality of the
Lagrangian distribution, which is a necessary condition for the simplification
of caustics to be possible, is also sufficient. We give applications of this
result to the arborealization program and to the study of nearby Lagrangian
homotopy spheres.
