[Paper Review] Constructing fold maps by surgery operations and their Reeb spaces

Naoki Kitazawa|arXiv (Cornell University)|Aug 23, 2015

Topological and Geometric Data Analysis17 references19 citations

TL;DR

This paper introduces and analyzes surgery operations—specifically M-bubbling and S-bubbling—on fold maps to construct new maps whose Reeb spaces have controlled homology groups. By iteratively modifying fold maps via these operations, the authors demonstrate how to systematically alter the homology of the Reeb space while preserving the underlying manifold structure, providing a powerful tool for classifying manifolds via fold map invariants.

ABSTRACT

In this paper, as a fundamental study on the theory of Morse functions and their higher dimensional versions or fold maps and applications to geometric theory of manifolds, which were started in 1950s by differential topologists such as Thom and Whitney and have been studied actively, we study algebraic and differential topological properties of certain fold maps and their source manifolds. More precisely, we investigate fold maps obtained by surgery operations to fundamental fold maps and especially, homology groups of Reeb spaces, which are defined as the space of all connected components of inverse images, often inheriting important invariants of manifolds such as homology groups and fundamental and important tools in studying manifolds. Studies of this paper are especially motivated by the stream of studies of fold maps satisfying good (differential) topological properties such as special generic maps, which were defined in 1970s and studied since 1990s by Saeki and Sakuma, and round fold maps, which were introduced by the author in 2012--2014, and their source manifolds. Moreover, constructions of generic maps by fundamental surgeries to investigate manifolds by using generic maps, studied by Kobayashi and Saeki etc., also have motivated the present study.

Motivation & Objective

• To develop a systematic method for constructing fold maps on smooth manifolds using surgery operations.
• To analyze the homological properties of Reeb spaces associated with fold maps, especially under surgery operations.
• To generalize known results on special generic and round fold maps by introducing new operations that modify Reeb space homology.
• To establish conditions under which the homology of the Reeb space can be precisely controlled through finite iterations of bubbling operations.
• To provide a framework for classifying manifolds via fold maps by relating topological invariants of the Reeb space to those of the source manifold.

Proposed method

• The paper employs M-bubbling and S-bubbling operations as fundamental surgery techniques on fold maps to modify their Reeb spaces.
• These operations are applied iteratively to a base fold map, with generating polyhedra chosen in regular value sets to control the resulting topology.
• The construction relies on embedding bouquets of manifolds (or spheres) into the Reeb space, ensuring compatibility with the map's structure.
• Homology groups of the Reeb space are modified by adding direct summands $ G_j $, with $ G_j $ being finitely generated modules over a PID $ R $.
• The operations are applied within open balls in connected components of the regular value set, preserving the smooth structure and codimension of the singular set.
• The method ensures that resulting inverse images of regular values are either connected sums or handle attachments, depending on the operation type.

Experimental results

Research questions

• RQ1 How can surgery operations on fold maps be used to systematically alter the homology groups of their Reeb spaces?
• RQ2 What conditions ensure that a finite sequence of M-bubbling or S-bubbling operations produces a fold map with a desired Reeb space homology?
• RQ3 Can the diffeomorphism type of the source manifold be preserved while modifying the Reeb space's homology via these operations?
• RQ4 To what extent can the Reeb space's homology be controlled by choosing specific generating polyhedra in the regular value set?
• RQ5 How do the resulting inverse images of regular values behave under different types of bubbling operations?

Key findings

• A finite iteration of M-bubbling operations can produce a fold map $ f' $ such that $ H_j(W_{f'}; R) o H_j(W_f; R) igoplus G_j $, where $ G_j $ is a finitely generated module over a PID $ R $.
• When $ G_j $ is generated by $ g_{n-k} $ copies of $ k $-dimensional spheres, the corresponding Reeb space homology is isomorphic to the direct sum with $ G_j $.
• For any $ j $, the homology group $ H_j(W_{f'}; R) $ can be made isomorphic to $ H_j(W_f; R) igoplus G_j $, provided $ G_0 $ is trivial and $ G_n $ is non-zero.
• The construction allows arbitrary choice of connected components of inverse images of regular values at each step, with specific constraints at M-bubbling steps.
• At M-bubbling steps, the inverse image splits into two components whose connected sum is the original component, while other steps allow handle attachment of index less than $ m-n $.
• The method is generalizable: the same result holds when the generating polyhedra are PL homeomorphic to bouquets of closed, connected, orientable manifolds of dimension less than $ n $, with homology isomorphic to that of spheres.