Geodesic
On the topology of stable Lagrangian maps with singularities of types $A$ and $D$
Izvestiya. Mathematics , Tome 79 (2015) no. 3, pp. 581-622.

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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.
Keywords: stable Lagrangian maps, multisingularities, adjacency index, Euler characteristic. 
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V. D. Sedykh. On the topology of stable Lagrangian maps with singularities of types $A$ and $D$. Izvestiya. Mathematics , Tome 79 (2015) no. 3, pp. 581-622. http://geodesic.mathdoc.fr/item/IM2_2015_79_3_a5/

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