商品簡介
《代數拓撲導論》講述了：This textbook is designed to introduce advanced undergraduate or beginning graduate students to algebraic topology as painlessly as possible. The principal topics treated are 2-dimensional manifolds, the fundamental group, and covering spaces, plus the group theory needed in these topics. The only prerequisites are some group theory, such as that normally contained in an undergraduate algebra course on the junior-senior level, and a one-semester undergraduate course in general topology.
The topics discussed in this book are standard in the sense that several well-known textbooks and treatises devote a few sections or a chapter to them. This, I believe, is the first textbook giving a straightforward treatment of these topics, stripped of all unnecessary definitions, terminology, etc., and with numerous examples and exercises, thus making them intelligible to advanced undergraduate students.
目次
CHAPTER ONE Two-Dimensional Manifolds
1 Introduction
2 Definition and examples of n-manifolds
3 Orientable vs. nonorientable manifolds
4 Examples of compact, connected 2-manifolds
5 Statement of the classification theorem for compact surfaces
6 Triangulations of compact surfaces
7 Proof of Theorem 5.1
8 The Euler characteristic of a surface
9 Manifolds with boundary
10 The classification of compact, connected 2-manifolds with boundary
11 The Euler characteristic of a bordered surface
12 Models of compact bordered surfaces in Euclidean 3-space
13 Remarks on noncompact surfaces
CHAPTER TWO The Fundamental Group
1 Introduction
2 Basic notation and terminology
3 Definition of the fundamental group of a space
4 The effect of a continuous mapping on the fundamental group
5 The fundamental group of a circle is infinite cyelic
6 Application: The Brouwer fixed-point theorem in dimension 2
7 The fundamental group of a product space
8 Homotopy type and homotopy equivalence of spaces
CHAPTER THREE Free Groups and Free Products of Groups
1 Introduction
2 The weak product of abelian groups
3 Free abelian groups
4 Free products of groups
5 Free groups
6 The presentation of groups by generators and relations
7 Universal mapping problems
CHAPTER FOUR Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces.Applica tions
1 Introduction
2 Statement and proof of the theorem of Seifert and Van Kampen
……
CHAPTER FIVE Covering Spaces
CHAPTER SIX The Fundamental Group and Covering Spaces of a Graph.Applications to Group Theory
CHAPTER SEVEN The Fundamental Group of Higher Dimensional Spaces
CHAPTER EIGHT Epilogue
APPENDIX A The Quotient Space or Identification Space Topology
Permutation Groups or Transformation Groups
Index
