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

Constructing Morse functions and their higher dimensional versions or foldmaps is fundamental, important and challenging in investigating the topologiesand the differentiable structures of differentiable manifolds via Morsefunctions, fold maps and more general generic maps. It is one of important andinteresting branches of the singularity theory of differentiable maps andapplications to geometry of manifolds. In this paper we present fold maps withinformation of cohomology rings of their Reeb spaces. Reeb spaces are definedas the spaces of all connected components of all preimages, and in suitablesituations inherit topological information such as homology groups andcohomology rings of the manifolds. Previously, the author demonstratedconstruction of fold maps in various cases : key methods are surgery operationsto manifolds and maps and in this paper, we present more useful surgeryoperations and by them we construct new fold maps. More precisely, fold mapswith singular value sets with crossings: the singular value set of a smooth mapis the image of the set of all singular points and note that for fold maps, theset of all singular points are closed submanifolds without boundaries and therestrictions to them are immersions of codimension 1.

Surgery operations to fold maps to construct fold maps whose restrictions to the singular sets may not be embeddings