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 $A_{\mu_1}^{\pm}\dotsb A_{\mu_p}^{\pm}$ 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.