Classifying immersions into \({\mathbb{R}}^ 4\) over stable maps of 3- manifolds into \({\mathbb{R}}^ 2\)
type: Item
instance of: scholarly article
title: Classifying immersions into \({\mathbb{R}}^ 4\) over stable maps of 3- manifolds into \({\mathbb{R}}^ 2\)
zbMATH Open document ID: 0567.57001
published in: Lecture Notes in Mathematics
publication date: 1985-01-01T00:00:00Z
review text: The first aim of this book is a study of stable maps in the sense of \textit{J. Mather} [Adv. Math. 4, 301-336 (1970; Zbl 0207.543)] of compact 3- dimensional manifolds into the plane. A stable map \(f: M^ 3\to {\mathbb{R}}^ 2\) allows 3 types of local singularities (definite and indefinite fold points and cusp points) and has to be normal on its singular set. The global geometry of a stable map \(f: M\to {\mathbb{R}}^ 2\) is investigated via its Stein factorization \(M\to^{q}W_ f\to^{\bar f}{\mathbb{R}}^ 2\), where \(W_ f\) is obtained by identifying points of M that belong to the same component of an f-fibre. In Chapter I, the local geometry of \(W_ f\) is studied and all local descriptions of \(W_ f\) are given. For orientable M, Chapter II develops semi-local canonical forms for the coordinate expressions of f near the singular set of f. In the final Chapter III, the following problem is addressed: Given a stable map \(f: M\to {\mathbb{R}}^ 2\) for M a compact orientable 3-manifold. How to classify f-regular immersions \(M\to {\mathbb{R}}^ 4\), i.e., immersions (f,h): \(M\to {\mathbb{R}}^ 2\times {\mathbb{R}}^ 2\), up to f-regular homotopy ? Here (f,h) and (f,h') are f-regular homotopic if there is a homotopy H joining h and h' such that \((f,H_ t)\) is an immersion for each t. The result, which of course is in terms of the decomposition of f into understandable pieces from the preceding chapters, is too involved to be stated here. It has to reflect the possible choices of h on each component of the non-singular part and on the singular arcs and compatibility conditions arising when piecing together. It is in the nature of the subject that most of this book is technical and difficult to read. Lots of instructive drawings, introductions and summaries to every chapter, and a guide to the abundant notation do help the reader considerably.
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 58C25
Mathematics Subject Classification ID: 58K99
Mathematics Subject Classification ID: 57R42
Mathematics Subject Classification ID: 57R45
zbMATH DE Number: 3905461
zbMATH Keywords: canonical coordinatized product neighborhood
zbMATH Keywords: stable maps
zbMATH Keywords: compact 3- dimensional manifolds
zbMATH Keywords: local singularities
zbMATH Keywords: Stein factorization
zbMATH Keywords: f- regular immersions
author: Harold I. Levine
reviewed by: Martin Raussen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0075066
OpenAlex ID: W428614550
DOI: 10.1007/BFB0075066
description: scientific article; zbMATH DE number 3905461