Constructing fold maps by surgery operations and their Reeb spaces

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.

31 pages, 9 figures. Theorem 8 added. Excuse me for making mistakes due to my carelessness

References in corpus (1)

• Round fold maps and the topologies and the differentiable structures of manifolds admitting explicit ones

Cited by in corpus (16)

• Notes on fold maps obtained by surgery operations and algebraic information of their Reeb spaces
• Notes on explicit smooth maps on 7-dimensional manifolds into the 4-dimensional Euclidean space
• Surgery operations to fold maps to construct fold maps whose restrictions to the singular sets may not be embeddings
• Surgery operations to fold maps to increase connected components of singular sets by two
• New observations on cohomology rings of Reeb spaces of explicit fold maps and manifolds admitting these maps
• Notes on explicit special generic maps into Eulidean spaces whose dimensions are greater than 4
• Realization problems of graphs as Reeb graphs of Morse functions with prescribed preimages
• Explicit remarks on the torsion subgroups of homology groups of Reeb spaces of explicit fold maps
• The images of special generic maps of an elementary class
• Global topologies of Reeb spaces of stable fold maps with non-trivial top homology groups
• The topologies and the differentiable structures of the images of special generic maps having simple structures
• New explicit construction of fold maps on general 7-dimensional closed and simply-connected spin manifolds
• A differential topological study of compact manifolds having simple structures
• Branched surfaces homeomorphic to Reeb spaces of simple fold maps
• 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-
• An elementary study on realizable changes of homology groups of Reeb spaces of fold maps by fundamental surgery operations