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

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 differential-topological properties 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 results for m-functions. For targets of dimension greater than one, we focus on the cases where the restrictions to the boundary are round fold maps or image simple fold maps, both defined by conditions on the singular value set. Then, we study the diffeomorphism types and Euler characteristics of manifolds admitting such maps. These results have applications to non-singular extensions of definite fold maps.