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＜電子ブック＞
Algebraic Topology : A First Course / by William Fulton
(Graduate Texts in Mathematics. ISSN:21975612 ; 153)

版	1st ed. 1995.
出版者	(New York, NY : Springer New York : Imprint: Springer)
出版年	1995
大きさ	XVIII, 430 p. 13 illus : online resource
著者標目	*Fulton, William author SpringerLink (Online service)
件　名	LCSH:Topology FREE:Topology
一般注記	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 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 HTTP:URL=https://doi.org/10.1007/978-1-4612-4180-5

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