Foreword: MASS and REU at Penn State University Preface 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
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