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Old and New Results on Density of Stable Mappings

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Handbook of Geometry and Topology of Singularities III

Abstract

Density of stable maps is the common thread of this paper. We review Whitney’s contribution to singularities of differentiable mappings and Thom-Mather theories on C∞ and C0-stability. Infinitesimal and algebraic methods are presented in order to prove Theorems A and B on density of proper stable and topologically stable mappings f : Nn → Pp. Theorem A states that the set of proper stable maps is dense in the set of all proper maps from N to P, if and only if the pair (n, p) is in nice dimensions, while Theorem B shows that density of topologically stable maps holds for any pair (n, p). A short review of results by du Plessis and Wall on the range in which proper smooth mappings are C1- stable is given. A Thom-Mather map is a topologically stable map f : N → P whose associated k-jet map jk f : N → P is transverse to the Thom-Mather stratification in Jk(N, P). We give a detailed description of Thom-Mather maps for pairs (n, p) in the boundary of the nice dimensions. The main open question on density of stable mappings is to determine the pairs (n, p) for which Lipschitz stable mappings are dense. We discuss recent results by Nguyen, Ruas and Trivedi on this subject, formulating conjectures for the density of Lipschitz stable mappings in the boundary of the nice dimensions. At the final section, Damon’s results relating \(\mathcal {A}\)-classification of map-germs and \(\mathcal {K}_{V}\) classification of sections of the discriminant of a stable unfolding of f are reviewed and open problems are discussed.

The analysis of the conditions for a map-germ to be finitely determined and of the degree of determinacy involves the most important of the local aspects of singularity theory.

C. T. C. Wall [ 108 ]

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Acknowledgements

This research was partially supported by CNPq grant # 305695/2019-3 and FAPESP, 2019/21181-0.

I am grateful to Jose Seade for the invitation to publish this work as a chapter of the “Handbook of Geometry and Topology of Singularities.”

Special thanks are due to David Trotman, David Mond and Raul Oset-Sinha for their valuable comments on preliminary versions of this paper, and to Débora Lopes for the help with the figures.

It is a pleasure to thank Nhan Nguyen ans Saurabh Trivedi for helpful discussions on Lipschitz stability of smooth mappings. I am also grateful to the referee for his careful reading and detailed remarks that improved very much the presentation of these notes.

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  1. Institute of Mathematical and Computer Sciences—USP, São Carlos, Brazil

    Maria Aparecida Soares Ruas

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  1. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México and, Unidad Mixta Internacional CNRS Laboratorio Solomon Lefschetz, Cuernavaca, Mexico

    José Luis Cisneros-Molina

  2. Centre de Mathématiques et Informatique, Universite d’Aix-Marseille, Marseille, France

    Lê Dũng Tráng

  3. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Mexico City, Mexico

    José Seade

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Ruas, M.A.S. (2022). Old and New Results on Density of Stable Mappings. In: Cisneros-Molina, J.L., Dũng Tráng, L., Seade, J. (eds) Handbook of Geometry and Topology of Singularities III. Springer, Cham. https://doi.org/10.1007/978-3-030-95760-5_1

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