On the complement to a real caustic germ of type $E_6$

We prove that the complement to the caustic of a stable Lagrangian map germ of type $E_6^{\pm }$ has seven connected components, six of which are contractible and one is homotopy equivalent to a circle. The inverse image of the noncontractible component under this map has three connected components. The restriction of the map to one of them is a two-sheeted covering; the restriction to each of the other two is a diffeomorphism.