Geodesic
Maps with prescribed Boardman singularities
Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, Tome 492 (2020), pp. 62-64 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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},
     year = {2020},
     volume = {492},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DANMA_2020_492_a12/}
}
TY  - JOUR
AU  - A. D. Ryabichev
TI  - Maps with prescribed Boardman singularities
JO  - Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
PY  - 2020
SP  - 62
EP  - 64
VL  - 492
UR  - http://geodesic.mathdoc.fr/item/DANMA_2020_492_a12/
LA  - ru
ID  - DANMA_2020_492_a12
ER  - 
%0 Journal Article
%A A. D. Ryabichev
%T Maps with prescribed Boardman singularities
%J Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ
%D 2020
%P 62-64
%V 492
%U http://geodesic.mathdoc.fr/item/DANMA_2020_492_a12/
%G ru
%F DANMA_2020_492_a12
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/

[1] Arnold V.I., Varchenko A.N., Gusein-Zade S.M., Osobennosti differentsiruemykh otobrazhenii, v. 1, Klassifikatsiya kriticheskikh tochek, kaustik i volnovykh frontov, Nauka, M., 1982, 304 pp.

[2] Boardman J.M., “Singularities of differentiable maps”, IHES Publ. Math., 33 (1967), 21–57 | DOI | MR | Zbl

[3] Whitney H., “Tangents to an Analytic Variety”, Ann. Math., 81:3 (1965), 496–549 | DOI | MR | Zbl

[4] Eliashberg Ya.M., “Ob osobennostyakh tipa skladki”, Izv. AN SSSR. Ser. matem., 34:5 (1970), 1110–1126 | Zbl

[5] Ryabichev A., Maps of manifolds of the same dimension with prescribed Thom-Boardman singularities, arXiv: 1810.10990

[6] Ryabichev A., “Eliashberg's $h$-principle and generic maps of surfaces with prescribed singular locus”, Topology and its Applications, 276 (2020) (to appear) | DOI | MR | Zbl