Surgery operations to fold maps to construct fold maps whose restrictions to the singular sets may not be embeddings

Abstract

Constructing Morse functions and their higher dimensional versions or fold maps is fundamental, important and challenging in investigating the topologies and the differentiable structures of differentiable manifolds via Morse functions, fold maps and more general generic maps. It is one of important and interesting branches of the singularity theory of differentiable maps and applications to geometry of manifolds. In this paper we present fold maps with information of cohomology rings of their Reeb spaces. Reeb spaces are defined as the spaces of all connected components of all preimages, and in suitable situations inherit topological information such as homology groups and cohomology rings of the manifolds. Previously, the author demonstrated construction of fold maps in various cases : key methods are surgery operations to manifolds and maps and in this paper, we present more useful surgery operations and by them we construct new fold maps. More precisely, fold maps with singular value sets with crossings: the singular value set of a smooth map is the image of the set of all singular points and note that for fold maps, the set of all singular points are closed submanifolds without boundaries and the restrictions to them are immersions of codimension 1.

18 pages, 3 figures, the title changed and various errors including crucial errors in main theorems corrected, this is submitted to a (refereed academic) journal

References in corpus (4)

• Round fold maps and the topologies and the differentiable structures of manifolds admitting explicit ones
• Standard Special Generic Maps of Homotopy Spheres into Euclidean Spaces
• Constructing fold maps by surgery operations and their Reeb spaces
• Notes on fold maps obtained by surgery operations and algebraic information of their Reeb spaces

Cited by in corpus (5)

• Surgery operations to fold maps to increase connected components of singular sets by two
• 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
• A differential topological study of compact manifolds having simple structures
• Branched surfaces homeomorphic to Reeb spaces of simple fold maps