Synopses & Reviews
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 differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously but economically, with minimal prerequisites and plenty of geometric intuition. A course on manifolds differs from most other introductory mathematics graduate courses in that the subject matter is often completely unfamiliar. Unlike algebra and analysis, which all math majors see as undergraduates, manifolds enter the curriculum much later. It is even possible to get through an entire undergraduate mathematics education without ever hearing the word "manifold." Yet manifolds are part of the basic vocabulary of modern mathematics, and students need to know them as intimately as they know the integers, the real numbers, Euclidean spaces, groups, rings, and fields. In his beautifully conceived introduction, the author motivates the technical developments to follow by explaining some of the roles manifolds play in diverse branches of mathematics and physics. Then he goes on to introduce the basics of general topology and continues with the fundamental group, covering spaces, and elementary homology theory. Manifolds are introduced early and used as the main examples throughout. John M. Lee is currently Professor of Mathematics at the University of Washington.
Review
This book is an introduction to manifolds on the beginning graduate level. It provides a readable text allowing every mathematics student to get a good knowledge of manifolds in the same way that most students come to know real numbers, Euclidean spaces, groups, etc. It starts by showing the role manifolds play in nearly every major branch of mathematics.
The book has 13 chapters and can be divided into five major sections. The first section, Chapters 2 through 4, is a brief and sufficient introduction to the ideas of general topology: topological spaces, their subspaces, products and quotients, connectedness and compactness.
The second section, Chapters 5 and 6, explores in detail the main examples that motivate the rest of the theory: simplicial complexes, 1- and 2-manifolds. It introduces simplicial complexes in both ways---first concretely, in Euclidean space, and then abstractly, as collections of finite vertex sets. Then it gives classification theorems for 1-manifolds and compact surfaces, essentially following the treatment in W. Massey's \ref[ Algebraic topology: an introduction, Reprint of the 1967 edition, Springer, New York, 1977
Synopsis
Manifolds play an important role in topology, geometry, complex analysis, algebra, and classical mechanics. Learning manifolds differs from most other introductory mathematics in that the subject matter is often completely unfamiliar. This introduction guides readers by explaining the roles manifolds play in diverse branches of mathematics and physics. The book begins with the basics of general topology and gently moves to manifolds, the fundamental group, and covering spaces.
Synopsis
This is an introductory text on manifolds, the technical detail of the definition and the applications in topology, geometry, complex analysis, algebra and classical mechanics.
Description
Includes bibliographical references (p. [359]-361) and index.
Table of Contents
Introduction.- General Topology.- New Spaces From Old.- Compactness and Connectedness.- Surfaces.- Homotopy and the Fundamental Group.- The Circle.- Some Group Theory.- Fundamental Groups of Surfaces.- Covering Spaces.- Classification of Covering Spaces.