Introduction to Topological Manifolds

John M. Lee (auth.)9780387950266, 9780387987590, 0387950265, 0387987592, 9780387227276

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.

Although this second edition has the same basic structure as the first edition, it has been extensively revised and clarified; not a single page has been left untouched. The major changes include a new introduction to CW complexes (replacing most of the material on simplicial complexes in Chapter 5); expanded treatments of manifolds with boundary, local compactness, group actions, and proper maps; and a new section on paracompactness.

This text is designed to be used for an introductory graduate course on the geometry and topology of manifolds. It should be accessible to any student who has completed a solid undergraduate degree in mathematics. The author’s book Introduction to Smooth Manifolds is meant to act as a sequel to this book.

Table of contents :
Front Matter….Pages i-xvii
Introduction….Pages 1-17
Topological Spaces….Pages 19-48
New Spaces from Old….Pages 49-84
Connectedness and Compactness….Pages 85-126
Cell Complexes….Pages 127-158
Compact Surfaces….Pages 159-182
Homotopy and the Fundamental Group….Pages 183-216
The Circle….Pages 217-231
Some Group Theory….Pages 233-250
The Seifert–Van Kampen Theorem….Pages 251-275
Covering Maps….Pages 277-305
Group Actions and Covering Maps….Pages 307-337
Homology….Pages 339-380
Back Matter….Pages 381-433