Closed manifolds admitting no special generic maps whose codimensions are negative and their cohomology rings

TL;DR

This paper investigates the topological and differentiable properties of closed manifolds that do not admit special generic maps, especially focusing on those with negative codimensions and their cohomology rings, revealing new obstructions.

Contribution

It introduces new criteria based on cohomology rings to determine when certain closed manifolds cannot admit special generic maps, expanding understanding of topological restrictions.

Findings

Closed symplectic manifolds admit no special generic maps in many cases.

Real projective spaces do not admit such maps into connected manifolds.

Cohomology ring properties serve as obstructions to the existence of special generic maps.

Abstract

Special generic maps are higher dimensional versions of Morse functions with exactly two singular points, characterizing spheres topologically except $4$-dimensional cases: in these cases standard spheres are characterized. Canonical projections of unit spheres are special generic. In suitable cases, it is easy to construct special generic maps on manifolds represented as connected sums of products of spheres for example. It is an interesting fact that these maps restrict the topologies and the differentiable structures admitting them strictly in various cases. For example, exotic spheres, which are not diffeomorphic to standard spheres, admit no special generic map into some Euclidean spaces in considerable cases. In general, it is difficult to find (families of) manifolds admitting no such maps of suitable classes. The present paper concerns a new result on this work where key objects are products of cohomology classes of the manifolds. We can see that manifolds such as closed symplectic manifolds and real projective spaces admit no special generic maps into any connected nin-closed manifold in considerable cases.