Regular homotopy classes of locally generic mappings

2003/10/14 by András Juhász, Andras Juhasz
Decision Sciences · Mathematics · #Advanced Topology and Set Theory #Functional Equations Stability Results #Fuzzy and Soft Set Theory #math.AT #math.GT #msc:57R42 #msc:57R45 #msc:58K30

paper · pdf · doi:10.1016/j.topol.2003.07.004

published as Topology Appl. 138 (2004), no. 1-3, 45--59 · 14 pages, 5 figures

openalex publication_date 2003/10/14 · arxiv created 2005/06/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we generalize the notion of regular homotopy of immersions of a closed connected n-manifold into R^2n-1 to locally generic mappings. The main result is that if n=2 then two mappings with singularities are regularly homotopic if and only if they have the same number of cross-cap (or Whitney-umbrella) singularities. As an application, we get a description of the path-components of the space of those immersions of a surface into R⁴ whose projections into R³ are locally generic.

Related

• Regular homotopy classes of singular maps
• Subgroups of the Mapping Class Group and Quadruple Points of Regular Homotopies
• Maps of manifolds of the same dimension with prescribed Thom-Boardman singularities
• Special generic maps and Gromoll filtration
• Regular homotopy classes of links of simple singularities and immersions associated with their Dynkin diagrams