دانلود کتاب Algebraic Topology: A First Course

This book introduces the important ideas of algebraic topology by emphasizing the relation of these ideas with other areas of mathematics. Rather than choosing one point of view of modern topology (homotropy theory, axiomatic homology, or differential topology, say) the author concentrates on concrete problems in spaces with a few dimensions, introducing only as much algebraic machinery as necessary for the problems encountered. This makes it possible to see a wider variety of important features in the subject than is common in introductory texts; it is also in harmony with the historical development of the subject. The book is aimed at students who do not necessarily intend on specializing in algebraic topology.Content Level » Lower undergraduateRelated subjects » Geometry & TopologyCoverS TitleAlgebraic Topology: A First CourseCopyright © 1995 Springer-Verlag ISBN 0-387-94327-7 (softcover) QA612F85 1995 514' 2—dc20 ISBN 0-387-94327-7 ISBN 3-540-94327-7 SPIN 10762531DedicationPrefaceContentsPART I: CALCULUS IN THE PLANE CHAPTER 1: Path Integrals 1a. Differential Forms and Path Integrals 1b. 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 PathsPART 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. SandwichesPART 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 Sc. The Jordan Curve Theorem 3d. 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 SpacesPART 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 Torus and Other Surfaces 8b. The Euler CharacteristicPART V: COHOMOLOGY AND HOMOLOGY, II CHAPTER 9: Holes and Integrals 9a. Multiply Connected Regions 9b. Integration over Continuous Paths and Chains 9c. Periods of Integrals 9d. Complex Integration CHAPTER 10: Mayer—Vietoris 10a. The Boundary Map 10b. Mayer—Vietoris for Homology 10c. Variations and Applications 10d. Mayer—Vietoris for CohomologyPART VI: COVERING SPACES AND FUNDAMENTAL GROUPS, I CHAPTER 11: Covering Spaces 11a. Definitions 11b. Lifting Paths and Homotopies 11c. G-Coverings 11d. Covering Transformations CHAPTER 12: The Fundamental Group 12a. Definitions and Basic Properties 12b. Homotopy 12c. Fundamental Group and HomologyPART VII: COVERING SPACES AND FUNDAMENTAL GROUPS, II CHAPTER 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 CHAPTER 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 GroupsPART VIII: COHOMOLOGY AND HOMOLOGY, III CHAPTER 15: Cohomology 15a. Patching Coverings and tech Cohomology 15b. Cech Cohomology and Homology 15c. De Rham Cohomology and Homology 15d. Proof of Mayer—Vietoris fo rDe Rham Cohomology CHAPTER 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 CocyclesPART IX: TOPOLOGY OF SURFACES CHAPTER 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 CHAPTER 18: Cohomology on Surfaces 18a. 1-Forms and Homology 18b. Integrals of 2-Forms 18d. De Rham Theory on SurfacesPART 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 21a. Spaces of Functions and 1-Forms 21b. Adeles 21c. Riemann—Roch 21d. The Abel—Jacobi TheoremPART 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 ComplexesAPPENDICES Conventions and Notation 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 TheoremHints and AnswersReferencesIndex of SymbolsIndex