卷绕数是拓扑学中最基本的不变量之一。它测量一个动点P绕一个不动点Q运动的次数，前提是P的运动路径不经过Q并且P的最终位置和它的起始位置相同。这个简单的想法有着深远的应用。通过本书的学习，读者将了解以下内容：卷绕数如何帮助我们证明每个多项式方程都有一个根（代数基本定理），保证通过单个平面切割对空间中三个对象进行公平划分（火腿三明治定理），解释为什么每个简单的闭曲线都有内部和外部（Jordan 曲线定理），将微积分与曲率和向量场的奇点联系起来（Hopf指数定理），允许从无穷中减去无穷并得到一个有限的答案（Toeplitz算子），推广给出关于矩阵群拓扑的一个基本且美丽的洞见（Bott周期性定理）。本书适合对卷绕数的概念及其在分析、微分几何和拓扑等数学领域中的应用感兴趣的本科生和研究生阅读。 本书涉及很多领域，但它以一种清晰而审慎的方式来表述，对于有所准备的大学生来说，这将是一本极好的读物。本书也是一项重要的研究，即一个直观的想法如何将人带入数学研究的深海。 —John McCleary, Mathematical Reviews 大学数学老师经常发现自己阅读了很多有关该主题的书。但即使对我们这些爱读书的人来说，当你读了大约十本线性代数书籍后（它们看起来都像是出自同一个模具），这个过程偶尔也会变得不那么吸引人了。因此，偶然发现一本真正独特的书是非常愉快的，它以一种特有的方式阐述了一个主题，并教给你一些以前不知道的东西。如果这本书在这方面还做得非常好，那就更好了，就像本书一样……Roe的写作风格简洁，但清晰而优雅；我读这本书的时候几乎能听到英国口音。这种清晰的写作风格和大量的附录使得本书更易于阅读。 —MAA Online

前辅文
Chapter 1. Prelude: Love, Hate, and Exponentials
  §1.1. Two sets of travelers
  §1.2. Winding around
  §1.3. The most important function in mathematics
  §1.4. Exercises
Chapter 2. Paths and Homotopies
  §2.1. Path connectedness
  §2.2. Homotopy
  §2.3. Homotopies and simple-connectivity
  §2.4. Exercises
Chapter 3. The Winding Number
  §3.1. Maps to the punctured plane
  §3.2. The winding number
  §3.3. Computing winding numbers
  §3.4. Smooth paths and loops
  §3.5. Counting roots via winding numbers
  §3.6. Exercises
Chapter 4. Topology of the Plane
  §4.1. Some classic theorems
  §4.2. The Jordan curve theorem I
  §4.3. The Jordan curve theorem II
  §4.4. Inside the Jordan curve
  §4.5. Exercises
Chapter 5. Integrals and the Winding Number
  §5.1. Differential forms and integration
  §5.2. Closed and exact forms
  §5.3. The winding number via integration
  §5.4. Homology
  §5.5. Cauchy’s theorem
  §5.6. A glimpse at higher dimensions
  §5.7. Exercises
Chapter 6. Vector Fields and the Rotation Number
  §6.1. The rotation number
  §6.2. Curvature and the rotation number
  §6.3. Vector fields and singularities
  §6.4. Vector fields and surfaces
  §6.5. Exercises
Chapter 7. The Winding Number in Functional Analysis
  §7.1. The Fredholm index
  §7.2. Atkinson’s theorem
  §7.3. Toeplitz operators
  §7.4. The Toeplitz index theorem
  §7.5. Exercises
Chapter 8. Coverings and the Fundamental Group
  §8.1. The fundamental group
  §8.2. Covering and lifting
  §8.3. Group actions
  §8.4. Examples
  §8.5. The Nielsen-Schreier theorem
  §8.6. An application to nonassociative algebra
  §8.7. Exercises
Chapter 9. Coda: The Bott Periodicity Theorem
  §9.1. Homotopy groups
  §9.2. The topology of the general linear group
Appendix A. Linear Algebra
  §A.1. Vector spaces
  §A.2. Basis and dimension
  §A.3. Linear transformations
  §A.4. Duality
  §A.5. Norms and inner products
  §A.6. Matrices and determinants
Appendix B. Metric Spaces
  §B.1. Metric spaces
  §B.2. Continuous functions
  §B.3. Compact spaces
  §B.4. Function spaces
Appendix C. Extension and Approximation Theorems
  §C.1. The Stone-Weierstrass theorem
  §C.2. The Tietze extension theorem
Appendix D. Measure Zero
  §D.1. Measure zero subsets of R and of S1
Appendix E. Calculus on Normed Spaces
  §E.1. Normed vector spaces
  §E.2. The derivative
  §E.3. Properties of the derivative
  §E.4. The inverse function theorem
Appendix F. Hilbert Space
  §F.1. Definition and examples
  §F.2. Orthogonality
  §F.3. Operators
Appendix G. Groups and Graphs
  §G.1. Equivalence relations
  §G.2. Groups
  §G.3. Homomorphisms
  §G.4. Graphs
Bibliography
Index