On planar caustics

Goryunov, VV and Gallagher, K (2016) On planar caustics. Journal of Knot Theory and Its Ramifications, 25 (12). p. 1642004.

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Abstract

We study local invariants of planar caustics, that is, invariants of Lagrangian maps from surfaces to R2 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 L of the Lagrangian maps. We 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, we use the discriminantal cycles to establish non-contractibility of certain loops in L.

Item Type:	Article
Uncontrolled Keywords:	Lagrangian map, caustic, local bifurcation, normal form local invariant, discriminantal cycle
Depositing User:	Symplectic Admin
Date Deposited:	06 Jul 2016 08:32
Last Modified:	19 Jan 2023 07:34
DOI:	10.1142/S0218216516420049
Related URLs:	Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3002116