Survey of apparent contours of stable maps between surfaces

Takahiro Yamamoto

Abstract

This is a survey paper about studies of the simplest shape of the apparent contour for stable maps between surfaces. Such studies first appeared in [10] then in [1], [3], [6], [20], [22]. Let $M$ be a connected and closed surface, $N$ a connected surface. For a stable map $\varphi: M\to N$, denote by $c(\varphi)$, $n(\varphi)$ and $i(\varphi)$ the numbers of cusps, nodes and singular set components of $\varphi$, respectively. For a $C^\infty$ map $\varphi_0 : M\to S^2$ into the sphere, we study the minimal pair $(i, c+n)$ and triples $(i,c,n)$, $(c,i,n)$, $(n,c,i)$ and $(i,n,c)$ among stable maps $M\to S^2$ homotopic to $\varphi_0$ with respect to the lexicographic order.

Article information

Dates
Revised: 26 March 2013
First available in Project Euclid: 19 October 2018

https://projecteuclid.org/ euclid.aspm/1539916277

Digital Object Identifier
doi:10.2969/aspm/06610013

Mathematical Reviews number (MathSciNet)
MR3382040

Zentralblatt MATH identifier
1360.57037

Keywords
Stable map cusp node

Citation

Yamamoto, Takahiro. Survey of apparent contours of stable maps between surfaces. Singularities in Geometry and Topology 2011, 13--29, Mathematical Society of Japan, Tokyo, Japan, 2015. doi:10.2969/aspm/06610013. https://projecteuclid.org/euclid.aspm/1539916277