Constructing fold maps by surgery operations and their Reeb spaces

2015/08/23 by Naoki Kitazawa, Kitazawa, Naoki · 2 citations
Computer Science · Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #K-Theory and Homology (math.KT) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1508.05630

openalex publication_date 2015/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Cited by

• Realizing a homology class of a compact manifold by a homology class of an explicit closed submanifold--a new approach to Thom's works on homology classes of submanifolds-
• Explicit remarks on the torsion subgroups of homology groups of Reeb spaces of explicit fold maps

Related

• Notes on fold maps obtained by surgery operations and algebraic information of their Reeb spaces
• An elementary study on realizable changes of homology groups of Reeb spaces of fold maps by fundamental surgery operations
• Surgery operations to fold maps to construct fold maps whose restrictions to the singular sets may not be embeddings
• Explicit remarks on the torsion subgroups of homology groups of Reeb spaces of explicit fold maps
• New observations on cohomology rings of Reeb spaces of explicit fold maps and manifolds admitting these maps