  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.
