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AuthorFulton, William. author
TitleAlgebraic Topology [electronic resource] : A First Course / by William Fulton
ImprintNew York, NY : Springer New York, 1995
Connect tohttp://dx.doi.org/10.1007/978-1-4612-4180-5
Descript XVIII, 430 p. 13 illus. online resource

SUMMARY

To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the reยญ 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, differยญ ential topology, etc.), we concentrate our attention on concrete probยญ lems in low dimensions, introducing only as much algebraic machinยญ 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 topolยญ ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical develยญ opment of the subject. What would we like a student to know after a first course in toยญ 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: underยญ standing the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; windยญ ing numbers and degrees of mappings, fixed-point theorems; appliยญ cations such as the Jordan curve theorem, invariance of domain; inยญ dices of vector fields and Euler characteristics; fundamental groups


CONTENT

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 Mayerโ{128}{148}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 Riemannโ{128}{148}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; Gaussโ{128}{153}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 Borsukโ{128}{153}s Theorem -- Hints and Answers -- References -- Index of Symbols


Mathematics Topology Mathematics Topology



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