Constructing fold maps by surgery operations and their Reeb spaces

TL;DR

This paper explores the algebraic and differential topology of fold maps created through surgery operations, focusing on the homology of Reeb spaces and their applications to understanding manifold structures.

Contribution

It introduces new methods for constructing fold maps via surgery and analyzes the homology of their Reeb spaces, advancing the understanding of manifold invariants.

Findings

Homology groups of Reeb spaces are affected by surgery operations.

Constructed fold maps exhibit specific topological properties.

Applications to the study of manifold invariants and classifications.

Abstract

In this paper, as a fundamental study on the theory of Morse functions and their higher dimensional versions or fold maps and applications to geometric theory of manifolds, which were started in 1950s by differential topologists such as Thom and Whitney and have been studied actively, we study algebraic and differential topological properties of certain fold maps and their source manifolds. More precisely, we investigate fold maps obtained by surgery operations to fundamental fold maps and especially, homology groups of Reeb spaces, which are defined as the space of all connected components of inverse images, often inheriting important invariants of manifolds such as homology groups and fundamental and important tools in studying manifolds. Studies of this paper are especially motivated by the stream of studies of fold maps satisfying good (differential) topological properties such as special generic maps, which were defined in 1970s and studied since 1990s by Saeki and Sakuma, and round fold maps, which were introduced by the author in 2012--2014, and their source manifolds. Moreover, constructions of generic maps by fundamental surgeries to investigate manifolds by using generic maps, studied by Kobayashi and Saeki etc., also have motivated the present study.