  In the singularity and differential topological theory of Morse functions and
higher dimensional versions or fold maps and application to algebraic and
differential topology of manifolds, constructing explicit fold maps and
investigating their source manifolds is fundamental, important and difficult.
The author has introduced surgery operations (bubbling operations) to fold
maps, motivated by studies of Kobayashi, Saeki etc. since 1990 and has
explicitly shown that homology groups of Reeb spaces of maps constructed by
iterations of these operations are flexible in several cases. Such operations
seem to be strong tools in construction of maps and precise studies of
manifolds. More precisely, the author has also noticed that the resulting
groups are represented as direct sums of the original homology groups and
suitable finitely generated commutative groups. The Reeb space of a map is the
space of all connected components of inverse images of the maps. Reeb spaces
inherit fundamental invariants of the manifolds such as homology groups etc.
much in simple cases as polyhedra whose dimensions are equal to those of the
target spaces. This paper is on a new explicit study of changes of homology
groups of Reeb spaces of fold maps by the surgery operations. We present
explicit changes obtained by an approach via elementary theory of sequences of
numbers and fundamental continuous or differentiable functions.
