Caustics of Lagrangian homotopy spheres with stably trivial Gauss map

TL;DR

This paper characterizes stably trivial Lagrangian homotopy spheres and shows they can be deformed so their caustics only have fold singularities, linking stable triviality to caustic simplification.

Contribution

It provides a geometric description of stably trivial elements in $ ext{pi}_n U_n/O_n$ and proves their representatives can be simplified to only fold tangencies via Hamiltonian isotopy.

Findings

All stably trivial elements admit fold-only tangency representatives.

Stable triviality of Lagrangian distributions is both necessary and sufficient for caustic simplification.

Applications to arborealization and Lagrangian homotopy spheres studies.

Abstract

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.