Manifolds admitting special generic maps and their nice generalized multisections

TL;DR

This paper demonstrates that manifolds with special generic maps also admit well-structured generalized multisections, linking the properties of these maps to manifold decompositions and topological structures.

Contribution

It establishes that manifolds admitting special generic maps can be decomposed via nice generalized multisections, extending the understanding of manifold structures and their relation to these maps.

Findings

Manifolds with special generic maps admit nice generalized multisections.

Special generic maps characterize certain spheres and influence manifold topology.

The work connects special generic maps with manifold decompositions like multisections.

Abstract

We show that manifolds admitting special generic maps also admit nice generalized multisections. Special generic maps are natural generalized versions of Morse functions with exactly two singular points on closed manifolds, characterizing spheres whose dimensions are not $4$ topologically and the $4$-dimensional unit sphere, and canonical projections of unit spheres. They are shown to restrict the differentiable structures of spheres etc. and topologies of more general manifolds strongly by Saeki, Sakuma etc., followed by Nishioka, Wrazidlo etc. and followed by the author. Some elementary or important manifolds also admit such maps. (Generalized) multisections of manifolds are nice decompositions of (compact) manifolds, generalizing so-called Heegaard splittings of $3$-dimensional manifolds. PL manifolds have been shown to have (generalized) multisections enjoying certain properties by Rubinstein and Tillmann.