Algebraic Topology

To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ ential topology, etc.), we concentrate our attention on concrete prob lems in low dimensions, introducing only as much algebraic machin ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel opment of the subject. What would we like a student to know after a first course in to pology (assuming we reject the answer: half of what one would like the student to know after a second course in topology)? Our answers to this have guided the choice of material, which includes: under standing the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; wind ing numbers and degrees of mappings, fixed-point theorems; appli cations such as the Jordan curve theorem, invariance of domain; in dices of vector fields and Euler characteristics; fundamental groups

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 Mayer Vietoris. - 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 Riemann Roch 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.