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.