I Calculus in the Plane. - 1 Path Integrals. - 2 Angles and Deformations. - II Winding Numbers. - 3 The Winding Number. - 4 Applications of Winding Numbers. - III Cohomology and Homology I. - 5 De Rham Cohomology and the Jordan Curve Theorem. - 6 Homology. - IV Vector Fields. - 7 Indices of Vector Fields. - 8 Vector Fields on Surfaces. - V Cohomology and Homology II. - 9 Holes and Integrals. - 10 MayerVietoris. - VI Covering Spaces and Fundamental Groups I. - 11 Covering Spaces. - 12 The Fundamental Group. - VII Covering Spaces and Fundamental Groups II. - 13 The Fundamental Group and Covering Spaces. - 14 The Van Kampen Theorem. - VIII Cohomology and Homology III. - 15 Cohomology. - 16 Variations. - IX Topology of Surfaces. - 17 The Topology of Surfaces. - 18 Cohomology on Surfaces. - X Riemann Surfaces. - 19 Riemann Surfaces. - 20 Riemann Surfaces and Algebraic Curves. - 21 The RiemannRoch Theorem. - XI Higher Dimensions. - 22 Toward Higher Dimensions. - 23 Higher Homology. - 24 Duality. - Appendices. - Appendix A Point Set Topology. - A1. Some Basic Notions in Topology. - A2. Connected Components. - A3. Patching. - A4. Lebesgue Lemma. - Appendix B Analysis. - B1. Results from Plane Calculus. - B2. Partition of Unity. - Appendix C Algebra. - C1. Linear Algebra. - C2. Groups; Free Abelian Groups. - C3. Polynomials; Gauss's Lemma. - Appendix D On Surfaces. - D1. Vector Fields on Plane Domains. - D2. Charts and Vector Fields. - D3. Differential Forms on a Surface. - Appendix E Proof of Borsuk's Theorem. - Hints and Answers. - References. - Index of Symbols. Language: English