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《代数拓扑导论(英文版)》介绍了：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.

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

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

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

……

CHAPTER 10 Covering Spaces

CHAPTER 11 Homotopy Groups

CHPATER 12 Cohomology

Bibliography

Notation

Index

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书籍介绍

《代数拓扑导论(英文版)》介绍了：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.