Number of Singularities of Stable Maps

Abstract

We first derive a simple Euler characteristic formula for simplicial maps, based on their fibers. As applications, we give several formulas relating the Euler characteristics and the numbers of singularities of smooth generic maps in explicit situations, among which are singular surfaces in 3-dimensional manifolds, Morse functions on surfaces, stable maps of 4-dimensional manifolds into 3-dimensional manifolds, etc.