Restrictions on special generic maps into ${\mathbb{R}}^5$ on $6$-dimensional or higher dimensional closed and simply-connected manifolds

TL;DR

This paper investigates restrictions on special generic maps from high-dimensional closed, simply-connected manifolds into 5-dimensional Euclidean space, revealing topological constraints and examples of manifolds admitting such maps.

Contribution

It establishes new topological restrictions on special generic maps for high-dimensional manifolds and identifies classes of manifolds that admit these maps, expanding understanding of their existence.

Findings

Certain non-diffeomorphic spheres do not admit such maps.

Connected sums of sphere products can admit special generic maps.

Cohomology ring analysis is crucial for studying map existence.

Abstract

The class of special generic maps is a natural class of smooth maps containing Morse functions on spheres with exactly two singular points and canonical projections of unit spheres. We find new restrictions on such maps on $6$-dimensional or higher dimensional closed and simply-connected manifolds into ${\mathbb{R}}^5$. Spheres which are not diffeomorphic to unit spheres do not admit such maps whose codimensions are negative in considerable cases. They restrict the homeomorphism and the diffeomorphism types of the manifolds in general. On the other hands, some elementary manifolds admit special generic maps into suitable Euclidean spaces: manifolds represented as connected sums of products of unit spheres are of such examples. This motivates us to study the (non-)existence of special generic maps on elementary manifolds such as projective spaces and some closed and simply-connected manifolds. For example, new explicit investigations of cohomology rings are keys in our new study.