  The Reeb space of a continuous map is the space of all (elements
representing) connected components of preimages endowed with the quotient
topology induced from the natural equivalence relation on the domain. These
objects are strong tools in (differential) topological theory of Morse
functions, fold maps, which are their higher dimensional variants, and so on:
they are in general polyhedra whose dimensions are same as those of the
targets. In suitable cases Reeb spaces inherit topological information such as
homology groups, cohomology rings, and so on, of the manifolds.
  This presents the following problem: what are global topologies of Reeb
spaces of these smooth maps of suitable classes like? The present paper
presents families of stable fold maps having Reeb spaces with non-trivial top
homology groups with their (co)homology groups (and rings). Related studies on
the global topologies from the viewpoints of the singularity theory of
differentiable maps and differential topology have been presented by various
researchers including the author. The author previously constructed families of
fold maps with Reeb spaces with non-trivial top homology groups and with good
topological properties. This paper presents new families, especially,
generalized situations of some known situations.

General Topology;Algebraic Topology