APPLICATIONS OF SINGULARITY THEORY TO GRAVITATIONAL LENSING .1. MULTIPLE LENS PLANES

The basic local and global features of stable multiple plane gravitati onal lens systems are investigated using tools from singularity theory . All stable multiple plane time-delay and lensing maps are classified , and the following global facts are proven under the weaker assumptio n of local stability. First, every locally stable multiple plane lensi ng map has an even number of cusps whether the associated deflector is singular or not. Second, for nonsingular deflectors the sum of the pr ojectivized rotation numbers of its caustics is zero, while for singul ar ones it is negative and even. Third, if the deflector has g point m asses on a single plane, then g is given by the formula g = - 1/2SIGMA (c)r(c), where r(c) is the projectivized rotation number of the critic al curve c and the sum runs through all critical curves. Fourth, expli cit counting formulas and bounds are found for the number of cusps for certain caustic networks. Finally, the latter yields that two point m asses on a single lens plane will generate at least six cusps. However , if the masses are put generically on separate lens planes, then ther e are at least eight cusps.