Global topologies of Reeb spaces of stable fold maps with non-trivial top homology groups

Abstract

The Reeb space of a continuous map is the space of all (elements representing) connected components of preimages endowed with the quotient topology induced from the natural equivalence relation on the domain. These objects are strong tools in (differential) topological theory of Morse functions, fold maps, which are their higher dimensional variants, and so on: they are in general polyhedra whose dimensions are same as those of the targets. In suitable cases Reeb spaces inherit topological information such as homology groups, cohomology rings, and so on, of the manifolds.This presents the following problem: what are global topologies of Reeb spaces of these smooth maps of suitable classes like? The present paper presents families of stable fold maps having Reeb spaces with non-trivial top homology groups with their (co)homology groups (and rings). Related studies on the global topologies from the viewpoints of the singularity theory of differentiable maps and differential topology have been presented by various researchers including the author. The author previously constructed families of stable fold maps with Reeb spaces with non-trivial top homology groups and with good topological properties and this paper presents new families.

Branched surfaces homeomorphic to Reeb spaces of simple fold maps

“…Remark 2. Some of arguments and results in the present paper can be generalized for branched manifolds, defined in [28], based on existing studies, and maps in the generalized cases. Generalizations are in some cases routine works and may be difficult in some cases.…”

“…Generalizations are in some cases routine works and may be difficult in some cases. [28] concentrates on global topologies of Reeb spaces in cases where the dimensions of the manifolds are general.…”

“…Remark 3. On global topologies of branched surfaces and maps born from SSNS fold maps, [28] explains existing related studies shortly. As presented there, [29] and [35] present or implicitly imply that branched surfaces resembling discs in homology groups (and fundamental groups) are regarded as Reeb spaces of simple fold maps.…”

“…The following class of 2-dimensional compact polyhedra is also presented in [28] for example. [9] presents a related classical theory.…”

“…Especially, construction of explicit smooth maps, which is important in the theory and has been difficult, and related works on global algebraic topological or differential topological properties of Reeb spaces by the author, have also motivated the present study. [17], [18], [20], [23], [25], [26], [28], and so on, are related studies. Studies concentrating more on manifolds admitting these maps are studied in [12], [13], [15], [16], [24], [27], and so on.…”