Topological properties of caustics in five-dimensional spaces

We give a list of universal linear relations between the Euler characteristics of manifolds consisting of multisingularities of a generic Lagrangian map into a five-dimensional space. From these relations it follows, in particular, that the numbers $D_5{A}_2,A_4{A}_3,A_4{A}_2^2$ of isolated self-intersection points of the corresponding types on any generic compact four-dimensional caustic are even. The numbers $D_4^{+}{A}_3+D_4^{-}{A}_3+E_6$, $D_4^{+}{A}_2^2+D_4^{-}{A}_2^2+\frac{1}{2}{A}_4{A}_3$ are even as well.