Global topologies of Reeb spaces of stable fold maps with non-trivial top homology groups

TL;DR

This paper investigates the global topologies of Reeb spaces of stable fold maps, demonstrating the existence of maps with Reeb spaces having non-trivial top homology groups and exploring their (co)homology structures.

Contribution

The paper introduces new families of stable fold maps with Reeb spaces exhibiting non-trivial top homology, extending previous constructions to more generalized cases.

Findings

Reeb spaces can have non-trivial top homology groups.

Constructed explicit examples of stable fold maps with complex Reeb space topologies.

Analyzed (co)homology rings of these Reeb spaces.

Abstract

The Reeb space of a continuous map is the space of all (elements representing) connected components of preimages endowed with the quotient topology induced from the natural equivalence relation on the domain. These objects are strong tools in (differential) topological theory of Morse functions, fold maps, which are their higher dimensional variants, and so on: they are in general polyhedra whose dimensions are same as those of the targets. In suitable cases Reeb spaces inherit topological information such as homology groups, cohomology rings, and so on, of the manifolds. This presents the following problem: what are global topologies of Reeb spaces of these smooth maps of suitable classes like? The present paper presents families of stable fold maps having Reeb spaces with non-trivial top homology groups with their (co)homology groups (and rings). Related studies on the global topologies from the viewpoints of the singularity theory of differentiable maps and differential topology have been presented by various researchers including the author. The author previously constructed families of fold maps with Reeb spaces with non-trivial top homology groups and with good topological properties. This paper presents new families, especially, generalized situations of some known situations.