English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Graph manifolds form important classes of $3$-dimensional closed andorientable manifolds. For example, {\it Seifert} manifolds are graph manifoldswhere hyperbolic manifolds are not.  In applying singularity theory of differentiable maps to understanding globaltopologies of manifolds, graph manifolds have been shown to be characterized asones admitting so-called simple fold maps into the plane of explicit classes bySaeki and the author. The present paper presents several related new results.  Fold maps are higher dimensional variants of Morse functions and simple onesform simple classes, generalizing the class of general Morse functions. Suchmaps into the plane on $3$-dimensional closed and orientable manifolds inducequotient maps to so-called simple polyhedra with no vertices, which are$2$-dimensional. This is also closely related to the theory of {\it shadows} of$3$-dimensional manifolds. We also discuss invariants for graph manifolds viaembeddability of these polyhedra in some $3$-dimensional manifolds.

Characterizing families of graph manifolds via suitable classes of  simple fold maps into the plane and embeddability of the Reeb spaces in some  3-dimensional manifolds