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

## TL;DR

This paper studies how homology groups of Reeb spaces of fold maps change under surgery operations, providing explicit descriptions and new insights into the topological flexibility of these maps.

## Contribution

It introduces a detailed analysis of homology group changes of Reeb spaces via elementary methods, expanding understanding of fold map modifications.

## Key findings

- Homology groups of Reeb spaces can be explicitly described after surgery operations.
- The changes are represented as direct sums involving original homology groups and finitely generated groups.
- Elementary sequence and function theory effectively analyze homology modifications.

## Abstract

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.