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.