代数拓扑 内容简介

本书是代数学基本观点的一个很好的展示。作者写这本书的想法来源于1955年他在芝加哥大学的演讲。从那时到现在代数学经历了很大的发展,该书的思想也是一直在更新,现在的这个版本是原版的修订版,称得上是一本真正的现代代数拓扑学。既可以作为教科书,也是一本很好的参考书。
本书分为三个主要部分,每部分包含三章。前三章都是在讲述基础群。**章给出其定义;第二章讲述覆盖空间;第三章发生器和关系,同时引进了多面体。四、五、六章都是在为下面章节研究同调理论做铺垫。第四章定义了同调;第五章涉及到更高层次的代数概念:上同调、上积,和上同调运算;第六章主要讲解拓扑流形。*后三章仔细研究了同调的概念。第七章介绍了同调群的基本概念;第八章将其应用于障碍理论;第九章给出了球体同调群的计算。每一个新概念的引入都会有应用实例来加深读者对它的理解。这些章节重点在于强调代数工具在几何中的应用。每章节后都有一些关于本章的练习。既有常规性的练习,又有部分是很具有激发性的,这些都可以帮助读者更好地了解本课程。
本书为全英文版。

代数拓扑 本书特色

IN THE MORE THAN TWENTY YEARS SINCE THE FIRST APPEARANCE OF Algebraic Topology the book has met with favorable response both in its use as a text and as a reference. It was the first comprehensive treatment of the fundamentals of the subject. Its continuing acceptance attests to the fact that its content and organization are still as timely as when it first appeared. Accord-ingly it has not been revised.

代数拓扑 目录

INTRODUCTION
1 Set theory
2 General topology
3 Group theory
4 Modules
5 Euclidean spaces
1 HOMOTOPy AND THE FUNDAMENTAL GROUP
1 Categories
2 Functors
3 Homotopy
4 Retraction and deforma
5 H spaces
6 Suspension
7 The fundamental groupoid
8 The fundamental group Exercises
2 COVERING SPACES AND FIHHATIONS
1 Covering protections
2 The homotopy lifting property
3 Relations with the fundamental group
4 The lifting problem
5 The classification of covering protections
6 Covering transformations
7 Fiber bundles
8 Fibrations Exercises
3 POLYBEDHA
1 Simplicial complexes
2 Linearity in simpltctal complexes
3 Subdivision
4 Simplicial approximation
5 Contiguity classes
6 The edge-path groupoid
7 Graphs
8 Examples and applications Exercises
4 HOMOLOGY
1 Chain complexes
2 Chain homotopy
3 The homology of simpltctal complexes
4 Singular homology
5 Exactness
6 Mayer-Vietorls sequences
7 Some applications of homology
8 Axiomatic characterization of homology Exercises
5 PRODUCTS
6 GENERAL COHOMOLOGY THEORY AND DUALITY
7 HOMOTOPY THEORY
8 OBSTRU CTION THEORY
9 SPECTRAL SEQUENCES AND HOMOTOPY GROUPS OF SPHERES
INDEX

代数拓扑 节选

EFACE TO THE SECOND
SPRINGER PRINTING
IN THE MORE THAN TWENTY YEARS SINCE THE FIRST APPEARANCE OF
Algebraic Topology the book has met with favorable response both in its use
as a text and as a reference. It was the first comprehensive treatment of the
fundamentals of the subject. Its continuing acceptance attests to the fact that
its content and organization are still as timely as when it first appeared. Accord-
ingly it has not been revised.
Many of the proofs and concepts first presented in the book have become
standard and are routinely incorporated in newer books on the subject. Despite
this, Algebraic Topology remains the best complete source for the material
which every young algebraic topologist should know. Springer-Verlag is to be
commended for its willingness to keep the book in print for future topologists.
For the current printing all of the misprints known to me have been cor-
rected and the .bibliography has been updated.
Berkeley, California Edwin H. Spanier
December 1989
PⅡIRFACE
THIS BOOK IS AN EXPOSITION OF THE FUNDAMENTAL IDEAS OF ALGEBRAIC
topology．1t is intended t0 be used both as a text and as a reference．Patticular
emphasis has been placed on aaturality，and the book might well have been
titled Functorial Topology，．The reader iS riot assumed to have prior knowledge
ofalgebraic topology，but he is assumed to know something of general topology
alld algebra and to be mathcmatically SOphisticated． Specinc prerequisite
material is brieHy summarized iIl the Introdnction．
sirice A lgebraic Topolgy is a text，the exposit／on in the eadier chapters
is a g00d deal slower than in the later chapters．The reader is exDected t0
develop facility for the subjectashe progresses，and accordingly，the further
he is in the b00k，the more he iS called upon to fill in details of prooffs．
Because it is alSO intended as a reference，some attempt has been made to
include basic concepts whetller ahey are used in the book or not．As a result,
there is more material than is usuallygiyen in courses on出e subject．
The material is organized into three main parts，each part being made up
0f three chapters．Each chapter is broken into several sectiOhS which treat
individual topics with some degree of thoroughness and are the basic organi-
zational units of the text. In the first three chapters the underlying theme is
the fundamental group. This is defined in Chapter One, applied in Chapter
Two in the study of covering spaces, and described by means of generators
and relations in Chapter Three, where polyhedra are introduced. The concept
of functor and its applicability to topology are stressed here to motivate
interest in the other functors of algebraic topology.
Chapters Four, Five, and Six are devoted to homology theory. Chapter
Four contains'the first definitions of homology, Chapter Five contains further
algebraic concepts such as cohomology, cup products, and cohomology oper-
ations, and Chapter Six contains a study of topological manifolds. With each
new concept introduced applications are presented to illustrate its utility:.
The last three chapters study homotopy theory. Basic facts about homo-
topy groups are considered in Chapter Seven, applications to obstruction
theory are presented in Chap