An Introduction to Algebraic Topology / by Joseph J. Rotman
(Graduate Texts in Mathematics ; 119)
| データ種別 | 電子ブック |
|---|---|
| 出版情報 | New York, NY : Springer New York , 1988 |
| 本文言語 | 英語 |
| 大きさ | XIV, 438 p : online resource |
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| 内容注記 | 0 Introduction Notation Brouwer Fixed Point Theorem Categories and Functors 1.Some Basic Topological Notions Homotopy Convexity, Contractibility, and Cones Paths and Path Connectedness 2 Simplexes Affine Spaces Affine Maps 3 The Fundamental Group The Fundamental Groupoid The Functor ?1 ?1(S1) 4 Singular Homology Holes and Green’s Theorem Free Abelian Groups The Singular Complex and Homology Functors Dimension Axiom and Compact Supports The Homotopy Axiom The Hurewicz Theorem 5 Long Exact Sequences The Category Comp Exact Homology Sequences Reduced Homology 6 Excision and Applications Excision and Mayer-Vietoris Homology of Spheres and Some Applications Barycentric Subdivision and the Proof of Excision More Applications to Euclidean Space 7 Simplicial Complexes Definitions Simplicial Approximation Abstract Simplicial Complexes Simplicial Homology Comparison with Singular Homology Calculations Fundamental Groups of Polyhedra The Seifert-van Kampen Theorem 8 CW Complexes Hausdorff Quotient Spaces Attaching Cells Homology and Attaching Cells CW Complexes Cellular Homology 9 Natural Transformations Definitions and Examples Eilenberg-Steenrod Axioms Chain Equivalences Acyclic Models Lefschetz Fixed Point Theorem Tensor Products Universal Coefficients Eilenberg-Zilber Theorem and the Künneth Formula 10 Covering Spaces Basic Properties Covering Transformations Existence Orbit Spaces 11 Homotopy Groups Function Spaces Group Objects and Cogroup Objects Loop Space and Suspension Homotopy Groups Exact Sequences Fibrations A Glimpse Ahead 12 Cohomology Differential Forms Cohomology Groups Universal Coefficients Theorems for Cohomology Cohomology Rings Computations and Applications Notation |
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| 一般注記 | There is a canard that every textbook of algebraic topology either ends with the definition of the Klein bottle or is a personal communication to J. H. C. Whitehead. Of course, this is false, as a glance at the books of Hilton and Wylie, Maunder, Munkres, and Schubert reveals. Still, the canard does reflect some truth. Too often one finds too much generality and too little attention to details. There are two types of obstacle for the student learning algebraic topology. The first is the formidable array of new techniques (e. g. , most students know very little homological algebra); the second obstacle is that the basic defini tions have been so abstracted that their geometric or analytic origins have been obscured. I have tried to overcome these barriers. In the first instance, new definitions are introduced only when needed (e. g. , homology with coeffi cients and cohomology are deferred until after the Eilenberg-Steenrod axioms have been verified for the three homology theories we treat-singular, sim plicial, and cellular). Moreover, many exercises are given to help the reader assimilate material. In the second instance, important definitions are often accompanied by an informal discussion describing their origins (e. g. , winding numbers are discussed before computing 1tl (Sl), Green's theorem occurs before defining homology, and differential forms appear before introducing cohomology). We assume that the reader has had a first course in point-set topology, but we do discuss quotient spaces, path connectedness, and function spaces |
| 著者標目 | *Rotman, Joseph J. author SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:Algebraic topology FREE:Mathematics FREE:Algebraic Topology |
| 分 類 | DC23:514.2 |
| 巻冊次 | ISBN:9781461245766 
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| ISBN | 9781461245766 |
| URL | http://dx.doi.org/10.1007/978-1-4612-4576-6 |
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