卷绕: 拓扑、几何和分析中的卷绕数（影印版）

样章

作者：

John Roe

定价：

135.00 元

版面字数：

488千字

开本：

16开

装帧形式：

精装

版次：

最新版次
印刷时间：

2023年

ISBN：

978-7-04-059314-3

物料号：

59314-00

出版时间：

2023-03-15

读者对象：

学术著作

一级分类：

自然科学

二级分类：

数学与统计

三级分类：

拓扑学

暂无

目录
- 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