Algebraic Topology A First Course

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Cover Title page Preface PART I: CALCULUS lN THE PLANE CHAPTER 1: Path IntegraIs 1a. DifferentiaI Forms and Path IntegraIs lb. When Are Path Integrals Independent of Path? 1c. A Criterion for Exactness CHAPTER 2:Angles and Deformations 2a. Angle Functions and Winding Numbers 2b. Reparametrizing and Deforming Paths 2c. Vector Fields and Fluid Flow PART II: WINDING NUMBERS CHAPTER 3: The Winding Number 3a. Definition of the Winding Number 3b. Homotopy and Reparametrization 3c. Varying the Point 3d. Degrees and Local Degrees CHAPTER 4: Applications of Winding Numbers 4a. The Fundamental Theorem of Algebra 4b. Fixed Points and Retractions 4c. Antipodes 4d. Sandwiches PART III: COHOMOLOGY AND HOMOLOGY, I CHAPTER 5: De Rham Cohomology and the Jordan Curve Theorem 5a. Definitions of the De Rham Groups 5b. The Coboundary Map 5c. The Jordan Curve Theorem 5d. Applications and Variations CHAPTER 6: Homology 6a. Chains, Cycles, and H₀U 6b. Boundaries, H₁U, and Winding Numbers 6c. Chains on Grids 6d. Maps and Homology 6e. The First Homology Group for General Spaces PART IV: VECTOR FIELDS CHAPTER 7: Indices of Vector Fields 7a. Vector Fields in the Plane 7b. Changing Coordinates 7c. Vector Fields on a Sphere CHAPTER 8: Vector Fields on Surfaces 8a. Vector Fields on a Toms and Other Surfaces 8b. The Euler Characteristic PART V: COHOMOLOGY AND HOMOLOGY, II CHAPTER 9: Holes and IntegraIs 9a. Multiply Connected Regions 9b. Integration over Continuous Paths and Chains 9c. Periods of IntegraIs 9d. Complex Integration CHAPTER 10: Mayer-Vietoris 10a. The Boundary Map 10b. Mayer-Vietoris for Homology 10c. Variations and Applications 10d. Mayer-Vietoris for Cohomology PART VI: COVERING SPACES AND FUNDAMENTAL GROUPS, I CHAPTER 11 Covering Spaces 11a. Definitions llb. Lifting Paths and Homotopies 11c. G-Coverings lld. Covering Transformations CHAPTER 12: The Fundamental Group l2a. Definitions and Basic Properties l2b. Homotopy l2c. Fundamental Group and Homology PART VII: COVERING SPACES AND FUNDAMENTAL GROUPS, II CHAPTER 13: The Fundamental Group and Covering Spaces 13a. Fundamental Group and Coverings l3b. Automorphisms of Coverings l3c. The Univers al Covering l3d. Coverings and Subgroups of the Fundamental Group CHAPTER 14: The Van Kampen Theorem l4a. G-Coverings from the Universal Covering l4b. Patching Coverings Together l4c. The Van Kampen Theorem l4d. Applications: Graphs and Free Groups PART VIII: COHOMOLOGY AND HOMOLOGY , III CHAPTER 15: Cohomology l5a. Patching Coverings and Cech Cohomology l5b. Cech Cohomology and Homology l5c. De Rham Cohomology and Homology l5d. Proof of Mayer-Vietoris for De Rham Cohomology CHAPTER 16: Variations l6a. The Orientation Covering l6b. Coverings from l-Forms l6c. Another Cohomology Group l6d. G-Sets and Coverings l6e. Coverings and Group Homomorphisms l6f. G-Coverings and Cocyc1es PART IX: TOPOLOGY OF SURFACES CHAPTER 17: The Topology of Surfaces l7a. Triangulation and Polygons with Sides Identified l7b. Classification of Compact Oriented Surfaces l7c. The Fundamental Group of a Surface CHAPTER 18: Cohomology on Surfaces l8a. l-Forms and Homology l8b. Integrals of 2-Forms l8c. Wedges and the Intersection Pairing l8d. De Rham Theory on Surfaces PART X: RIEMANN SURFACES CHAPTER 19: Riemann Surfaces 19a. Riemann Surfaces and Analytic Mappings 19b. Branched Coverings 19c. The Riemann-Hurwitz Formula CHAPTER 20: Riemann Surfaces and Algebraic Curves 20a. The Riemann Surface of an Algebraic Curve 20b. Meromorphic Functions on a Riemann Surface 20c. Holomorphic and Meromorphic 1-Forms 20d. Riemann's Bilinear Relations and the Jacobian 20e. Elliptic and Hyperelliptic Curves CHAPTER 21: The Riemann-Roch Theorem 2la. Spaces of Functions and l-Forms 2lb. Adeles 2lc. Riemann-Roch 2ld. The Abel-Jacobi Theorem PART XI: HIGHER DIMENSIONS CHAPTER 22: Toward Higher Dimensions 22a. Holes and Forms in 3-Space 22b. Knots 22c. Higher Homotopy Groups 22d. Higher De Rham Cohomology 22e. Cohomology with Compact Supports CHAPTER 23: Higher Homology 23a. Homology Groups 23b. Mayer-Vietoris for Homology 23c. Spheres and Degree 23d. Generalized Jordan Curve Theorem CHAPTER 24: Duality 24a. Two Lemmas from Homological Algebra 24b. Homology and De Rham Cohomology 24c. Cohomology and Cohomology with Compact Supports 24d. Simplicial Complexes APPENDICES APPENDIX A: Point Set Topology Al. Some Basic Notions in Topology A2. Connected Components A3. Patching A4. Lebesgue Lemma APPENDIX B: Analysis BI. Results from Plane Calculus B2. Partition of Unity APPENDIX C: Algebra CI. Linear Algebra C2. Groups; Free Abelian Groups C3. Polynomials; Gauss's Lemma APPENDIX D: On Surfaces Dl. Vector Fields on Plane Domains D2. Charts and Vector Fields D3. DifferentiaI Forms on a Surface APPENDIX E: Proof of Borsuk's Theorem Hints and Answers References Index of Symbols Index