Mapping Degree Theory
About this Title
Enrique Outerelo, Universidad Complutense de Madrid, Madrid, Spain and Jesús M. Ruiz, Universidad Complutense de Madrid, Madrid, Spain
Publication: Graduate Studies in Mathematics
Publication Year 2009: Volume 108
ISBNs: 978-0-8218-4915-6 (print); 978-1-4704-1171-8 (online)
MathSciNet review: MR2566906
MSC: Primary 55M25; Secondary 01A55, 01A60, 57R35
This textbook treats the classical parts of mapping degree theory, with a detailed account of its history traced back to the first half of the 18th century. After a historical first chapter, the remaining four chapters develop the mathematics. An effort is made to use only elementary methods, resulting in a self-contained presentation. Even so, the book arrives at some truly outstanding theorems: the classification of homotopy classes for spheres and the Poincaré-Hopf Index Theorem, as well as the proofs of the original formulations by Cauchy, Poincaré, and others.
Although the mapping degree theory you will discover in this book is a classical subject, the treatment is refreshing for its simple and direct style. The straightforward exposition is accented by the appearance of several uncommon topics: tubular neighborhoods without metrics, differences between class 1 and class 2 mappings, Jordan Separation with neither compactness nor cohomology, explicit constructions of homotopy classes of spheres, and the direct computation of the Hopf invariant of the first Hopf fibration.
The book is suitable for a one-semester graduate course. There are 180 exercises and problems of different scope and difficulty.
Graduate students interested in topology, particularly differential topology.
Table of Contents
- Chapter 1. History
- Chapter 2. Manifolds
- Chapter 3. The Brouwer-Kronecker degree
- Chapter 4. Degree theory in Euclidean spaces
- Chapter 5. The Hopf Theorems