微分拓扑是每个人都应该了解的理论。本书主要介绍了微分拓扑学的相关理论，通过对机器人手臂的介绍引入课程。全书共八章，包括微分拓扑简介、光滑映射、切线空间、常规值、向量丛、向量丛的结构、可积性和走向全球的局部现象。本书首先讨论了流形、切线空间、余切空间，其次讨论了丛的相关知识，最后自然地以切线和余切丛的讨论而告终。本书是一本适合具有一定数学水平的学生使用的教科书，内容由浅入深，适合高等院校师生、研究生及数学爱好者参考阅读。

Preface
1 Introduction
1.1 A Robot's Arm
1.2 The Configuration Space of Two Electrons
1.3 State Spaces and Fiber Bundles
1.4 Further Examples
1.5 Compact Surfaces
1.6 Higher Dimensions
2 Smooth Manifolds
2.1 Topological Manifolds
2.2 Smooth Structures
2.3 Maximal Atlases
2.4 Smooth Maps
2.5 Submanifolds
2.6 Products and Sums
3 The Tangent Space
3.1 Germs
3.2 Smooth Bump Functions
3.3 The Tangent Space
3.4 The Cotangent Space
3.5 Derivations
4 Regular Values
4.1 The Rank
4.2 The Inverse Function Theorem
4.3 The Rank Theorem
4.4 Regular Values
4.5 Transversality
4.6 Sard's Theorem
4.7 Immersions and Imbeddings
5 Vector Bundles
5.1 Topological Vector Bundles
5.2 Transition Functions
5.3 Smooth Vector Bundles
5.4 Pre-vector Bundles
5.5 The Tangent Bundle
5.6 The Cotangent Bundle
6 Constructions on Vector Bundles
6.1 Subbundles and Restrictions
6.2 The Induced Bundle
6.3 Whitney Sum of Bundles
6.4 Linear Algebra on Bundles
6.5 Normal Bundles
6.6 Riemannian Metrics
6.7 Orientations
6.8 The Generalized Gauss Map
7 Integrability
7.1 Flows and Velocity Fields
7.2 Integrability: Compact Case
7.3 Local Flows and Integrability
7.4 Second-Order Differential Equations
8 Local Phenomena that Go Global
8.1 Refinements of Covers
8.2 Partition of Unity
8.3 Global Properties of Smooth Vector Bundles
8.4 An Introduction to Morse Theory
8.5 Ehresmann's Fibration Theorem
Appendix A Point Set Topology
A.1 Topologies: Open and Closed Sets
A.2 Continuous Maps
A.3 Bases for Topologies
A.4 Separation
A.5 Subspaces
A.6 Quotient Spaces
A.7 Compact Spaces
A.8 Product Spaces
A.9 Connected Spaces
A.10 Set-Theoretical Stuff
Appendix B Hints or Solutions to the Exercises
References
Index