● Preface
1 Introduction
What Are Manifolds？
Why Study Manifolds？
2 Topological Spaces
Topologies
Convergence and Continuity
Hausdorff Spaces
Bases and Countability
Manifolds
Problems
3 New Spaces from Old
Subspaces
Product Spaces
Disjoint Union Spaces
Quotient Spaces
Adjunction Spaces
Topological Groups and Group Actions
Problems
4 Connectedness and Compactness
Connectedness
Compactness
Local Compactness
Paracompactness
Proper Maps
Problems
5 Cell Complexes
Cell Complexes and CW Complexes
Topological Properties of Cw Complexes
Classification of 1-Dimensional Manifold
Simpli Complexes
Problems
6 Compact Surfaces
Surfaces
Connected Sums of Surfaces
Polygonal Presentations of Surfaces
The Classification Theorem
The Euler Characteristic
Orientability
Problems
7 Homotopy and the Fundamental Group
Homotopy
The Fundamental Group
Homomorphisms Induced by Continuous Maps
Homotopy Equivalence
Higher Homotopy Groups
Categories and Functors
Problems
8 The Circle
Lifting Properties of the Circle
The Fundamental Group of the Circle
Degree Theory for the Circle
Problems
9 Some Group Theory
Free Products
Free Groups
Presentations of Groups
Free Abelian Groups
Problems
10 The Seifert-Van Kampen Theorem
Statement of the Theorem
Applications
Fundamental Groups of Compact Surfaces
Proof of the Seifert-Van Kampen Theorem
Problems
11 Covering Maps
Definitions and Basic Properties
The General Lifting Problem
The Monodromy Action
Covering Homomorphisms
The Universal Covering Space
Problems
12 Group Actions and Covering Maps
The Automorphism Group of a Covering
Ouotients by Group Actions
The Classification Theorem
Proper Group Actions
Problems
13 Homology
Singular Homology Groups
Homotopy Invariance
Homology and the Fundamental Grour
The Mayer-Vietoris Theorem
Homology of Spheres
Homology of CW Complexes
Cohomology
Problems
Appendix A: Review of Set Theory
Basic Concepts
Cartesian Products,Relations,and Functions
Number Systems and Cardinality
Indexed Families
Appendix B: Review of Metric Spaces
Euclidean Spaces
Metrics
Continuity and Convergence
Appendix C: Review of Group Theory
Basic Definitions
Cosets and Quotient Groups
Cyclic Groups
References
Notation Index
Subject Index
内容简介
本书作者是美国华盛顿大学教授，具有丰富的教学经验，他在华盛顿大学和哈佛大学教授流形课程已有15年之久。书中论述了流形理论中所需的拓扑学基本概念，特别是微分几何、代数几何和相关领域。线和曲面；同伦和基本群论；圆和球；群论；Seifert-Van Kampen定理；覆盖空间；覆盖类别；同调。
