On submersions with definite folds of manifolds with boundary into Euclidean spaces

TL;DR

This paper investigates the properties and classifications of submersions with definite folds on manifolds with boundary into Euclidean spaces, focusing on their topological and differential properties.

Contribution

It provides new restrictions on the diffeomorphism types of source manifolds and analyzes their Euler characteristics under various fold map conditions.

Findings

Restrictions on diffeomorphism types of source manifolds when target is R.

Analysis of Euler characteristics for manifolds with boundary admitting such maps.

Application to non-singular extensions of definite fold maps.

Abstract

Submersions with definite folds are submersions on manifolds with boundary whose restrictions to the boundary are definite fold maps. In this paper, we study the properties from the viewpoint of differential topology of manifolds with boundary admitting such maps into Euclidean spaces. When the target is $\mathbb{R}$, we obtain restrictions on the diffeomorphism types of the source manifolds by using previous results of Hajduk and Borodzik--N\'emethi--Ranicki. For Euclidean spaces with general dimensions, we consider submersions with definite folds whose restrictions to the boundary are round fold maps or image simple fold maps, both defined by imposing conditions on the singular point set. Then, we study the diffeomorphism types and Euler characteristics of the manifolds admitting such maps. These results are also applied to the study of non-singular extensions of definite fold maps.