Smooth maps like special generic maps

TL;DR

This paper introduces SGL maps, a new class of smooth maps that generalize special generic maps, to explore their algebraic and differential topological properties and their implications for manifold structures.

Contribution

The paper defines SGL maps as a broader class of smooth maps extending special generic maps, and studies their algebraic and differential topological properties.

Findings

SGL maps generalize special generic maps.

They impose restrictions on manifold topologies.

The study broadens understanding of smooth map classes.

Abstract

In our paper, we introduce special-generic-like maps or SGL maps as smooth maps and study their several algebraic topological and differential topological properties. The new class generalize the class of so-called special generic maps. Special generic maps are smooth maps which are locally projections or the product maps of Morse functions and the identity maps on disks. Morse functions with exactly two singular points on spheres or Morse functions in Reeb's theorem are simplest examples. Special generic maps and the manifolds of their domains have been studied well. Their structures are simple and this help us to study explicitly. As important properties, they have been shown to restrict the topologies and the differentiable structures of the manifolds strongly by Saeki and Sakuma, followed by Nishioka, Wrazidlo and the author. To cover wider classes of manifolds as the domains, the author previously introduced a class generalizing the class of special generic maps and smaller than our class: simply generalized special generic maps.