內容簡介
《代數拓撲導論(英文版)》介紹了：ThereisacanardthateverytextbookofalgebraictopologyeitherendswiththedefinitionoftheKleinbottleorisapersonalcommunicationto.I.H.C.Whitehead.Ofcourse,thisisfalse,asagianceatthebooksofHiltonandWylie,Maunder,Munkres,andSchubertreveals.Still,thecanarddoesreflectsometruth.Toooftenonefindstoomuchgeneralityandtoolittleattentiontodetails.
目錄
Preface
To the Reader
CHAPTER 0 Introduction
Notation
Brouwer Fixed Point Theorem
Categories and Functors
CHAPTER 1 Some Basic Topological Notions
Homotopy
Convexity, Contractibility, and Cones
Paths and Path Connectedness
CHAPTER 2 Simplexes
Affine Spaces
Aftine Maps
CHAPTER 3 The Fundamental Group
The Fundamental Groupoid
The Functor π
π1(S1)
CHAPTER 4 Singular Homology
Holesand Green's Theorem
Free Abelian Groups
The Singularcomplex and Homology Functors
Dimensionaxiomand Compact Supports
The Homotopy Axiom
The Hurewicz Theorem
CHAPTER 5 Long Exact Sequences
The Category Comp
Exact Homology Sequences
Reduced Homology
CHAPTER 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 Space
CHAPTER 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
CHAPTER 8 CW Complexes
Hausdorff Quotient Spaces
Attaching Calls
Homology and Attaching Cells
CW Complexes
Cellular Homology
CHAPTER 9 Natural Transformations
Definitions and Examples
Eilenberg-Steenrod Axioms
Chain Equivalences
Acyclic Models
Lefschetz Fixed Point Theorem
Tensor Products
Universal Coefficients
Eilcnberg-Zilber Theorem and the Kunneth Formula
CHAPTER 10
Covering Spaces
Basic Properties
Covering Transformations
Existence
Orbit Spaces
CHAPTER 11
Homotopy Groups
Function Spaces
Group Objects and Cogroup Objects
Loop Space and Suspension
Homotopy Groups
Exact SequenCes
Fihrations
AglimpseAhead
CHAPTER 12
Cohomology
Differential Forms
Cohomology Groups
Universal Coefficients Theorems for Cohomology
Cohomology Rings
Computations and Applications
Bibliography
Notation
Index
