Classifying immersions into \({\mathbb{R}}^ 4\) over stable maps of 3- manifolds into \({\mathbb{R}}^ 2\)
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.
canonical coordinatized product neighborhood
scientific article; zbMATH DE number 3905461