On the non-existence of special generic maps on complex projective spaces

TL;DR

This paper proves that complex projective spaces cannot admit special generic maps, extending previous results and providing a complete classification for these spaces in the context of Morse functions and related mappings.

Contribution

The paper establishes the non-existence of special generic maps on complex projective spaces, offering a comprehensive result that extends prior partial findings.

Findings

Complex projective spaces do not admit special generic maps.

The result generalizes previous non-existence proofs for real projective spaces.

Provides a complete classification for complex projective spaces regarding special generic maps.

Abstract

We prove the non-existence of special generic maps on complex projective space as our extended new result. Simplest special generic maps are Morse functions with exactly two singular points on spheres, or Morse functions in Reeb's theorem, and canonical projections of unit spheres. Manifolds represented as connected sums of products of manifolds diffeomorphic to unit spheres admit such maps in considerable cases. Real and complex projective spaces have been shown to admit no such maps in most cases by the author. This gives a complete answer for complex projective spaces as a corollary to a more general result, which is also our main result.