[Paper Review] Structures of cobordism-like modules induced from generic maps of codimension -2

Naoki Kitazawa|arXiv (Cornell University)|Jan 14, 2019

Homotopy and Cohomology in Algebraic Topology6 references4 citations

TL;DR

This paper investigates the algebraic structures of cobordism-like modules induced by generic smooth maps of codimension -2, focusing on Reeb-triangulable maps. By constructing explicit Morse and fold maps via handle attachments and gluing techniques, the authors demonstrate that these modules exhibit high flexibility, showing that any element in the module can be realized as the inverse image structure of such a map. The key contribution is a structural flexibility theorem for these modules in codimension -2.

ABSTRACT

The Reeb space of a smooth map whose codimension is minus is the space defined as the space of all connected components of inverse images. For generic maps such as Morse functions and their higher dimensional versions, they are polyhedra whose dimensions are equal to those of the target manifolds and which have simplicial structures compatible with (the canonical) simplicial structures of the source and the target manifolds, and in considerable cases they inherit fundamental and important invariants of source manifolds. In fact, Reeb spaces are fundamentall tools in the algebraic and differential topological theory of generic maps or the global singularity theory. As one of studies of global topological properties of Reeb spaces, Hiratuka and Saeki showed in 2013 that for generic maps or more precisely, maps compatible with simplicial structures of the manifolds, inducing simplicial structures on the Reeb spaces and having connected components of inverse images of regular values being not null-cobordant, the top-dimensional homology groups with appropriate coefficient rings of the Reeb spaces do not vanish. Later the author extended this theorem: the author has introduced cobordism-like groups based on adjacent relations of connected components of inverse images of regular values and shown a similar theorem.

Motivation & Objective

• To investigate the algebraic structure of cobordism-like modules induced by generic maps of codimension -2.
• To extend previous results on Reeb spaces and null-cobordism to a broader class of maps with negative codimension.
• To demonstrate that the module structures are flexible, meaning any element can be realized as an inverse image structure of a suitable map.
• To generalize earlier results on oriented and unoriented cobordism invariants via new constructions of smooth maps.
• To establish a framework for realizing arbitrary elements of the module through explicit geometric and topological operations on manifolds.

Proposed method

• Utilizes Reeb-triangulable maps—generalizations of Morse functions and fold maps—whose Reeb spaces are compatible with PL structures.
• Constructs local Morse and fold maps on manifolds using handle attachments (1- and 2-handles) to modify boundary components and achieve desired inverse image structures.
• Employs gluing techniques to combine local maps into global maps over a closed manifold $N$, ensuring the source manifold remains connected.
• Applies trivial bundle extensions over the complement of local map images to extend maps globally while preserving the inverse image structure.
• Uses product constructions with the identity on $S^{n-1}$ to generalize 1-dimensional results to higher-dimensional targets ($n > 1$).
• Relies on a key proposition (Proposition 1) that allows the realization of any generator in the module via appropriate boundary configurations and handle attachments.

Experimental results

Research questions

• RQ1 Can any element in the cobordism-like module of codimension -2 be realized as the inverse image structure of a generic smooth map?
• RQ2 What geometric and topological operations are sufficient to construct such maps with prescribed inverse image components?
• RQ3 How does the flexibility of the module structure manifest in the context of Reeb-triangulable maps and fold maps?
• RQ4 What are the limitations of realizing elements in the non-oriented or real projective case, and why does the obstruction arise?
• RQ5 To what extent can the module structure be controlled via handle attachments and gluing of local maps?

Key findings

• The cobordism-like module for codimension -2 maps is flexible: any element in the module can be realized as the inverse image structure of a Reeb-triangulable map.
• Explicit constructions via handle attachments (1- and 2-handles) allow transformation of boundary components into any desired configuration of connected, closed, oriented surfaces.
• The global map can be constructed over a closed manifold $N$ by gluing local Morse or fold maps and extending trivially over the complement, preserving the desired inverse image structure.
• For $n > 1$, the construction generalizes via product maps with the identity on $S^{n-1}$, ensuring the image contains a standard $n$-disc neighborhood of the singular value set.
• The obstruction to realizing elements in the non-oriented case (e.g., in ${ mf N}_2({f R})$) arises from non-null-cobordant manifolds like ${f R}P^2$, which cannot be realized as inverse images in such constructions.
• The proof relies on a key proposition showing that any generator can be represented as a combination of spheres with positive and negative coefficients, realizable via symmetric local maps and boundary gluing.