On planar caustics (Q2835370)

From the introduction: The authors study local invariants of planar caustics, that is, invariants of Lagrangian maps from surfaces to \(\mathbb R^2\) whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of the caustics. Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space \(\mathcal L\) of the Lagrangian maps.NEWLINENEWLINEThe authors obtain a description of the spaces of the discriminantal cycles (possibly non-trivial) for the Lagrangian maps of an arbitrary surface, both for the integer and mod 2 coefficients. It is shown that all integer local invariants of caustics of Lagrangian maps without corank 2 points are essentially exhausted by the numbers of various singular points of the caustics and the Ohmoto-Aicardi linking invariant of ordinary maps. As an application, the authors use the discriminantal cycles to establish non-contractibility of certain loops in \(\mathcal L\).