New observations on cohomology rings of Reeb spaces of explicit fold maps and manifolds admitting these maps

TL;DR

This paper presents new observations on the explicit construction of fold maps and the calculation of homology and cohomology rings of their Reeb spaces, advancing understanding in algebraic and differential topology.

Contribution

The paper introduces systematic methods for constructing explicit fold maps and computing the algebraic invariants of their Reeb spaces, providing new insights into their topological properties.

Findings

Explicit fold maps constructed systematically.

Homology groups of Reeb spaces calculated.

Cohomology rings of Reeb spaces analyzed.

Abstract

As a branch of algebraic and differential topology of manifolds, the theory of Morse functions and their higher dimensional versions or fold maps and its application to algebraic and differential topology of manifolds is fundamental, important and interesting. This paper is on explicit construction of fold maps and homology groups and cohomology rings of their Reeb spaces: they are defined as the spaces of all connected components of preimages of the maps, and in suitable situations inherit some topological information such as homology groups and cohomology rings of the manifolds. Explicit construction of the maps is a fundamental and difficult task even on manifolds which are not so complicated. The author has constructed explicit fold maps systematically and performed several calculations of homology groups and cohomology rings of the Reeb spaces. This paper concerns new observations on this task.