Algebraic Topology

A First Course
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William Fulton
680 g
235x155x24 mm
153, Graduate Texts in Mathematics

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
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I Calculus in the Plane.- 1 Path Integrals.- 1a. Differential Forms and Path Integrals.- 1b. When Are Path Integrals Independent of Path?.- 1c. A Criterion for Exactness.- 2 Angles and Deformations.- 2a. Angle Functions and Winding Numbers.- 2b. Reparametrizing and Deforming Paths.- 2c. Vector Fields and Fluid Flow.- II Winding Numbers.- 3 The Winding Number.- 3a. Definition of the Winding Number.- 3b. Homotopy and Reparametrization.- 3c. Varying the Point.- 3d. Degrees and Local Degrees.- 4 Applications of Winding Numbers.- 4a. The Fundamental Theorem of Algebra.- 4b. Fixed Points and Retractions.- 4c. Antipodes.- 4d. Sandwiches.- III Cohomology and Homology, I.- 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.- 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.- IV Vector Fields.- 7 Indices of Vector Fields.- 7a. Vector Fields in the Plane.- 7b. Changing Coordinates.- 7c. Vector Fields on a Sphere.- 8 Vector Fields on Surfaces.- 8a. Vector Fields on a Torus and Other Surfaces.- 8b. The Euler Characteristic.- V Cohomology and Homology, II.- 9 Holes and Integrals.- 9a. Multiply Connected Regions.- 9b. Integration over Continuous Paths and Chains.- 9c. Periods of Integrals.- 9d. Complex Integration.- 10 Mayer-Vietoris.- 10a. The Boundary Map.- 10b. Mayer-Vietoris for Homology.- 10c. Variations and Applications.- 10d. Mayer-Vietoris for Cohomology.- VI Covering Spaces and Fundamental Groups, I.- 11 Covering Spaces.- 11a. Definitions.- 11b. Lifting Paths and Homotopies.- 11c. G-Coverings.- 11d. Covering Transformations.- 12 The Fundamental Group.- 12a. Definitions and Basic Properties.- 12b. Homotopy.- 12c. Fundamental Group and Homology.- VII Covering Spaces and Fundamental Groups, II.- 13 The Fundamental Group and Covering Spaces.- 13a. Fundamental Group and Coverings.- 13b. Automorphisms of Coverings.- 13c. The Universal Covering.- 13d. Coverings and Subgroups of the Fundamental Group.- 14 The Van Kampen Theorem.- 14a. G-Coverings from the Universal Covering.- 14b. Patching Coverings Together.- 14c. The Van Kampen Theorem.- 14d. Applications: Graphs and Free Groups.- VIII Cohomology and Homology, III.- 15 Cohomology.- 15a. Patching Coverings and ?ech Cohomology.- 15b. ?ech Cohomology and Homology.- 15c. De Rham Cohomology and Homology.- 15d. Proof of Mayer-Vietoris for De Rham Cohomology.- 16 Variations.- 16a. The Orientation Covering.- 16b. Coverings from 1-Forms.- 16c. Another Cohomology Group.- 16d. G-Sets and Coverings.- 16e. Coverings and Group Homomorphisms.- 16f. G-Coverings and Cocycles.- IX Topology of Surfaces.- 17 The Topology of Surfaces.- 17a. Triangulation and Polygons with Sides Identified.- 17b. Classification of Compact Oriented Surfaces.- 17c. The Fundamental Group of a Surface.- 18 Cohomology on Surfaces.- 18a. 1-Forms and Homology.- 18b. Integrals of 2-Forms.- 18c. Wedges and the Intersection Pairing.- 18d. De Rham Theory on Surfaces.- X Riemann Surfaces.- 19 Riemann Surfaces.- 19a. Riemann Surfaces and Analytic Mappings.- 19b. Branched Coverings.- 19c. The Riemann-Hurwitz Formula.- 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.- 21 The Riemann-Roch Theorem.- 21a. Spaces of Functions and 1-Forms.- 21b. Adeles.- 21c. Riemann-Roch.- 21d. The Abel-Jacobi Theorem.- XI Higher Dimensions.- 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.- 23 Higher Homology.- 23a. Homology Groups.- 23b. Mayer-Vietoris for Homology.- 23c. Spheres and Degree.- 23d. Generalized Jordan Curve Theorem.- 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.- 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.

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