Geodesic
Maps with prescribed Boardman singularities
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 492 (2020), pp. 62-64

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In this paper we extend Y. Eliashberg's theorem on the maps with fold type singularities to arbitrary Thom-Boardman singularities. Namely, we state a necessary and sufficient condition for a continuous map of smooth manifolds of the same dimension to be homotopic to a generic map with a prescribed Thom-Boardman singularity $\Sigma^I$ at each point. In dimensions 2 and 3 we rephrase this condition in terms of the homology classes of the given singular loci and the characteristic classes of the manifolds.
Keywords: Thom-Boardman singularities, folds, cusps, $h$-principle. 
@article{DANMA_2020_492_a12,
     author = {A. D. Ryabichev},
     title = {Maps with prescribed {Boardman} singularities},
     journal = {Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleni\^a},
     pages = {62--64},
     publisher = {mathdoc},
     volume = {492},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_492_a12/}
}
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A. D. Ryabichev. Maps with prescribed Boardman singularities. Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 492 (2020), pp. 62-64. http://geodesic.mathdoc.fr/item/DANMA_2020_492_a12/