Lee Introduction to Topological Manifolds

Erscheinungsjahr 2006
ISBN: 978-0-387-22727-6
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 392 Seiten, Web PDF

Reihe: Graduate Texts in Mathematics

This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of di?erential geometry, algebraic topology, and related ?elds. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition. Here at the University of Washington, for example, this text is used for the ?rst third of a year-long course on the geometry and topology of manifolds; the remaining two-thirds focuses on smooth manifolds. Therearemanysuperbtextsongeneralandalgebraictopologyavailable. Why add another one to the catalog? The answer lies in my particular visionofgraduateeducation—itismy(admittedlybiased)beliefthatevery serious student of mathematics needs to know manifolds intimately, in the same way that most students come to know the integers, the real numbers, Euclidean spaces, groups, rings, and ?elds. Manifolds play a role in nearly every major branch of mathematics (as I illustrate in Chapter 1), and specialists in many ?elds ?nd themselves using concepts and terminology fromtopologyandmanifoldtheoryonadailybasis. Manifoldsarethuspart of the basic vocabulary of mathematics, and need to be part of the basic graduate education. The ?rst steps must be topological, and are embodied in this book; in most cases, they should be complemented by material on smooth manifolds, vector ?elds, di?erential forms, and the like. (After all, few of the really interesting applications of manifold theory are possible without using tools from calculus.

Zielgruppe

Graduate

Autoren/Hrsg.

Lee, John M.

Inhaltsverzeichnis

Topological Spaces.- New Spaces from Old.- Connectedness and Compactness.- Simplicial Complexes.- Curves and Surfaces.- Homotopy and the Fundamental Group.- Circles and Spheres.- Some Group Theory.- The Seifert-Van Kampen Theorem.- Covering Spaces.- Classification of Coverings.- Homology.