A new proof of a theorem of Dutertre and Fukui on Morin singularities

TL;DR

This paper presents a new proof of a theorem by Dutertre and Fukui concerning Morin singularities, utilizing Morse theory for manifolds with boundary to offer an alternative approach for the case when the target is Euclidean space.

Contribution

It introduces a novel proof of Dutertre-Fukui's theorem on Morin singularities using Morse theory, expanding the toolkit for analyzing singular sets in differential topology.

Findings

New proof of Dutertre-Fukui's theorem for N=ℝ^n

Application of Morse theory to study singular sets

Alternative approach to existing topological results

Abstract

In [2], N.Dutertre and T. Fukui used Viro's integral calculus to study the topology of stable maps $f:M\rightarrow N$ between two smooth manifolds $M$ and $N$. They also discussed several applications to Morin maps. In particular, in Theorem 6.2 [2], they show an equality relating the Euler characteristic of a compact manifold $M$ and the Euler characteristic of the singular sets of a Morin map defined on $M$. In this paper we show how Morse theory for manifolds with boundary can be applied to the study of the singular sets of a Morin map in order to obtain a new proof of Dutertre-Fukui's Theorem when $N=\mathbb{R}^n$.