On the topology of stable Lagrangian maps with singularities of types $A$ and $D$

We study the topology of adjacencies of multisingularities in the image of a stable Lagrangian map with singularities of types $A_{\mu }^{\pm }$ and $D_{\mu }^{\pm }$. In particular, we prove that each connected component of the manifold of multisingularities of any fixed type $A_{{\mu }_1}^{\pm }\cdots A_{{\mu }_p}^{\pm }$ for a germ of the image of a Lagrangian map with a monosingularity of type $D_{\mu }^{\pm }$ is either contractible or homotopy equivalent to a circle. We calculate the number of connected components of each kind for all types of multisingularities. As a corollary, we obtain new conditions for the coexistence of Lagrangian singularities.