## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Singularities in Geometry and Topology 2011, V. Blanlœil and O. Saeki, eds. (Tokyo: Mathematical Society of Japan, 2015), 13 - 29

### Survey of apparent contours of stable maps between surfaces

#### 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**

Received: 29 April 2012

Revised: 26 March 2013

First available in Project Euclid:
19 October 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.2969/aspm/06610013

**Mathematical Reviews number (MathSciNet)**

MR3382040

**Zentralblatt MATH identifier**

1360.57037

**Subjects**

Primary: 57R45: Singularities of differentiable mappings

Secondary: 58K15: Topological properties of mappings 57R35: Differentiable mappings

**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