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

An elementary study on realizable changes of homology groups of Reeb spaces of fold maps by fundamental surgery operations