New families of manifolds with similar cohomology rings admitting special generic maps

TL;DR

This paper explores new families of manifolds with similar cohomology rings that admit special generic maps, revealing connections between their topological structures and the existence of such maps.

Contribution

It introduces new manifolds with similar cohomology rings and clarifies their relation to the existence of special generic maps, expanding previous classifications.

Findings

Identified new families of manifolds with similar cohomology rings.

Established a link between cohomology ring structure and the existence of special generic maps.

Extended previous classifications of manifolds admitting special generic maps.

Abstract

As Reeb's theorem shows, Morse functions with exactly two singular points on closed manifolds are very simple and important. They characterize spheres whose dimensions are not $4$ topologically and the $4$-dimensional unit sphere. Special generic maps are generalized versions of these maps. Canonical projections of unit spheres are special generic. Studies of Saeki and Sakuma since the 1990s, followed by Nishioka and Wrazidlo, show that the differentiable structures of the spheres and the homology groups of the manifolds (in several classes) are restricted. We see special generic maps are attractive. Our paper studies the cohomology rings of manifolds admitting such maps. As our new result, we find a new family of manifolds whose cohomology rings are similar and find that the (non-)existence of special generic maps are closely related to the topologies. More explicitly, we have previously found related families and our new manifolds add to these discoveries.