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

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.

arXiv comment21 pages, some presentations revised, There are 6 theorems as new results in this version, this version is submitted to a refereed journal, mistakes etc. are due to my carelessness

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)

• Notes on explicit special generic maps into Eulidean spaces whose dimensions are greater than 4
• 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