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Foreword£ºMASS and REU at Penn State University Preface Chapter 1£®Prelude£º 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 ¢ñ 4£®3£®The Jordan curve theorem ¢ò 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 Chapter 9£®Coda£º The Bott Periodicity Theorem Appendix A£®Linear Algebra Appendix B£®Metric Spaces Appendix C£®Extension and Approximation Theorems Appendix D£®Measure Zero Appendix E£®Calculus on Normed Spaces Appendix F£®Hilbert Space Appendix G£®Groups and Graphs Bibliography Index Foreword£º MASS and REU at Penn State University Preface Chapter 1£®Prelude£º 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 Chapter 9£®Coda£º The Bott Periodicity Theorem Appendix A£®Linear Algebra Appendix B£®Metric Spaces Appendix C£®Extension and Approximation Theorems Appendix D£®Measure Zero Appendix E£®Calculus on Normed Spaces Appendix F£®Hilbert Space Appendix G£®Groups and Graphs Bibliography Index