On planar caustics

Victor Goryunov (University of Liverpool)

Frank Adams 2,

We study local invariants of planar caustics, that is, invariants of Lagrangian maps from surfaces to R^2 whose increments in generic homotopies are determined entirely by diffeomorphism types of local bifurcations of their critical value sets (= caustics). Such invariants are dual to trivial codimension 1 cycles supported on the discriminant in the space L of the Lagrangian maps. We obtain a description of the space of integer discriminantal cycles (possibly non-trivial) for the Lagrangian maps of an arbitrary surface. For the majority of these cycles we find homotopy-independent interpretations which guarantee the triviality required. As an application, we show that some of the discriminantal cycles are non-trivial, and establish with their help non-contractibility of certain loops in L. A crash course on Lagrangian maps and their local description will be included.

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