Complements of caustics of real function singularities

Abstract

We study the topology of complements of caustics of function singularities of low codimensions, in particular 1) complete the enumeration of connected components of the complements of caustics of {\em simple} (in the sense of V.Arnold) singularities, in particular find the numbers of these components for the last two classes, $E_7$ and $E_8$, remaining unknown after the works of R.Thom, V.Arnold and V.Sedykh; 2) realize all these components for simple singularities by explicitly constructed functions, and realize their one-dimensional homology and cohomology groups by cycles and cocycles; 3) prove that (in contrast to the case of simple singularities) for some parabolic singularities the two-dimensional homology groups of the complements of their caustics are nontrivial.