Characterizing families of graph manifolds via suitable classes of simple fold maps into the plane and embeddability of Reeb spaces in some 3dimensional manifolds
Abstract
Graph manifolds form important classes of 3dimensional closed and orientable manifolds. For example, Seifert manifolds are graph manifolds where hyperbolic manifolds are not. In applying singularity theory of differentiable maps to understanding global topologies of manifolds, graph manifolds have been shown to be characterized as ones admitting socalled simple fold maps into the plane of explicit classes by Saeki and the author. The present paper presents several new results of this type. Fold maps are higher dimensional variants of Morse functions and simple ones form simple classes, generalizing the class of general Morse functions. Such maps into the plane on $3$dimensional closed and orientable manifolds induce quotient maps to socalled simple polyhedra with no vertices, which are 2dimensional. This is also closely related to the theory of shadows of 3dimensional manifolds. We also discuss invariants for graph manifolds via embeddability of these polyhedra in some 3dimensional manifolds.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.08629
 Bibcode:
 2021arXiv210708629K
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  General Topology
 EPrint:
 16 pages, title changed, a sketch of a proof of Theorem 1 (3) added (essentially a main theorem of arxiv:2105.00974), several expositions revised, this will be improved before submission to a refereed journal