Proofs of the non-existence of special generic maps on the $3$-dimensional complex projective space

TL;DR

This paper proves that the 3-dimensional complex projective space cannot admit special generic maps, extending understanding of the limitations of such maps on certain closed, simply-connected manifolds.

Contribution

It establishes the non-existence of special generic maps on 3-dimensional complex projective space using multiple methods, filling a gap in the understanding of these maps on low-dimensional manifolds.

Findings

Proves non-existence of special generic maps on the 3D complex projective space

Extends previous results on restrictions of special generic maps to spheres

Uses multiple methods to establish non-existence

Abstract

We prove the non-existence of special generic maps on $3$-dimensional complex projective space as our new result and a corollary by several methods. Special generic maps are generalizations of Morse functions with exactly two singular points on spheres and canonical projections of unit spheres are special generic. Our paper focuses on such maps on closed and simply-connected manifolds of classes containing the $3$-dimensional complex projective space. The differentiable structures of spheres admitting special generic maps are known to be restricted strongly. Special generic maps on closed and simply-connected manifolds and projective spaces have been studied by various people including the author. The (non-)existence and construction are main problems. Studies on such maps on closed and simply-connected manifolds whose dimensions are greater than $5$ have been difficult.