代數(shù)拓?fù)鋵?dǎo)論

前言

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 .I.H.C. Whitehead. Of course, this is false, as a giance 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.Tbere 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 definitions 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 coefficents and cohomology are deferred until after the EilenbergSteenrod axioms have been verified for the three homology theories we treat——singular, simplicial, 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, Green's theorem occurs before defining homology, and differential forms appear before introducing cohomology）.

內(nèi)容概要

　　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 .I.H.C. Whitehead. Of course， this is false， as a giance 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.

作者簡(jiǎn)介

作者：(美國(guó))羅曼

書籍目錄

PrefaceTo the ReaderCHAPTER 0 Introduction  Notation  Brouwer Fixed Point Theorem  Categories and FunctorsCHAPTER 1 Some Basic Topological Notions  Homotopy  Convexity, Contractibility, and Cones  Paths and Path ConnectednessCHAPTER 2 Simplexes  Affine Spaces  Aftine MapsCHAPTER 3 The Fundamental Group  The Fundamental Groupoid  The Functor π  π1(S1)CHAPTER 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 TheoremCHAPTER 5 Long Exact Sequences  The Category Comp  Exact Homology Sequences  Reduced HomologyCHAPTER 6 Excision and Applications  Excision and Mayer-Vietoris  Homology of Spheres and Some Applications  Barycentric Subdivision and the Proof of Excision  Moxe Applications to Euclidean SpaceCHAPTER 7 Simplicial Complexes  Definitions  Simplicial Approximation  Abstract Simplicial Complexes  Simplicial Homology  Comparison with Singular Homology  Calculations  Fundamental Groups of Polyhedra  The Seifert-van Kampen TheoremCHAPTER 8 CW Complexes  Hausdorff Quotient Spaces  Attaching Calls  Homology and Attaching Cells  CW Complexes  Cellular HomologyCHAPTER 9 Natural Transformations  Definitions and Examples  Eilenberg-Steenrod Axioms  ……CHAPTER 10 Covering SpacesCHAPTER 11 Homotopy GroupsCHPATER 12 CohomologyBibliographyNotationIndex

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