目錄
Preface
Notation and Terminology
CHAPTER Ⅰ 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
References
CHAPTER Ⅱ 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 Cyclic
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
References
CHAPTER Ⅲ 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
References
CHAPTER Ⅳ Seifert and Van Kampen Theorem on the Fundamental Group of the Union of Two Spaces.Applications
1.Introduction
2.Statement and Proof of the Theorem of Seifert and Van Kampen
3.First Application of Theorem 2.1
4.Second Application of Theorem 2.1
5.Structure of the Fundamental Group of a Compact Surface
6.Application to Knot Theory
7.Proof of Lemma 2.4
References
CHAPTER Ⅴ Covering Spaces
1.Introduction
2.Definition and Some Examples of Covering Spaces
3.Lifting of Paths to a Covering Space
4.The Fundamental Group of a Covering Space
5.Lifting of Arbitrary Maps to a Covering Space
6.Homomorphisms and Automorphisms of Covering Spaces
7.The Action of the Group π(X,x) on the Set p-1 (x)
8.Regular Covering Spaces and Quotient Spaces
9.Application: The Borsuk-Ulam Theorem for the 2-Sphere
10.The Existence Theorem for Covering Spaces References
CHAPTER Ⅵ Background and Motivation for Homology Theory
1.Introduction
2.Summary of Some of the Basic Properties of Homology Theory
3.Some Examples of Problems which Motivated the Development of Homology Theory in the Nineteenth Century References
CHAPTER Ⅶ Definitions and Basic Properties of Homology Theory
1.Introduction
2.Definition of Cubical Singular Homology Groups
3.The Homomorphism Induced by a Continuous Map
4.The Homotopy Property of the Induced Homomorphisms
5.The Exact Homology Sequence of a Pair
6.The Main Properties of Relative Homology Groups
7.The Subdivision of Singular Cubes and the Proof of Theorem 6.4
CHAPTER Ⅷ Determination of the Homology Groups of Certain Spaces: Applications and Further Properties of Homology Theory
1.Introduction
2.Homology Groups of Cells and Spheres—Applications
3.Homology of Finite Graphs
4.Homology of Compact Surfaces
5.The Mayer-Vietoris Exact Sequence
6.The Jordan-Brouwer Separation Theorem and lnvariance of Domain
7.The Relation between the Fundamental Group and the First Homology Group
References
CHAPTER Ⅸ Homology of CW-Complexes
1.Introduction
2.Adjoining Cells to a Space
3.CW-Complexes
4.The Homology Groups of a CW-Complex
5.Incidence Numbers and Orientations of Cells
6.Regular CW-Complexes
7.Determination of Incidence Numbers for a Regular Cell Complex
8.Homology Groups of a Pseudomanifold
References
CHAPTER Ⅹ Homology with Arbitrary Coefficient Groups
1.Introduction
2.Chain Complexes
3.Definition and Basic Properties of Homology with Arbitrary Coefficients
4.Intuitive Geometric Picture of a Cycle with Coefficients in G
5.Coefficient Homomorphisms and Coefficient Exact Sequences
6.The Universal Coefficient Theorem
7.Further Properties of Homology with Arbitrary Coefficients
References
CHAPTER ? The Homology of Product Spaces
1.Introduction
2.The Product of CW-Complexes and the Tensor Product of Chain Complexes
3.The Singular Chain Complex of a Product Space
4.The Homology of the Tensor Product of Chain Complexes (The Kiinneth Theorem)
5.Proof of the Eilenberg-Zilber Theorem
6.Formulas for the Homology Groups of Product Spaces
References
CHAPTER ? Cohomology Theory
1.Introduction
2.Definition of Cohomology Groups-Proofs of the Basic Properties
3.Coefficient Homomorphisms and the Bockstein Operator in Cohomology
4.The Universal Coefficient Theorem for Cohomology Groups
5.Geometric Interpretation of Cochains,Cocycles,etc.
6.Proof of the Excision Property;the Mayer-Vietoris Sequence
References
CHAPTER ⅩⅢ Products in Homology and Cohomology
1.Introduction
2.The Inner Product
3.An Overall View of the Various Products
4.Extension of the Definition of the Various Products to Relative Homology and Cohomology Groups
5.Associativity,Commutativity,and Existence of a Unit of the Various Products
6.Digression: The Exact Sequence of a Triple or a Triad
7.Behavior of Products with Respect to the Boundary and Coboundary Operator of a Pair
8.Relations Involving the Inner Product
9.Cup and Cap Products in a Product Space
10.Remarks on the Coefficients for the Various Products-The Cohomology Ring
11.The Cohomology of Product Spaces (The K?nneth Theorem for Cohomology)
References
CHAPTER ⅩⅣ Duality Theorems for the Homology of Manifolds
1.Introduction
2.Orientability and the Existence of Orientations for Manifolds
3.Cohomology with Compact Supports
4.Statement and Proof of the Poincar? Duality Theorem
5.Applications of the Poincar? Duality Theorem to Compact Manifolds
6.The Alexander Duality Theorem
7.Duality Theorems for Manifolds with Boundary
8.Appendix: Proof of Two Lemmas about Cap Products
References
CHAPTER ⅩⅤ Cup Products in Projective Spaces and Applications of Cup Products
1.Introduction
2.The Projective Spaces
3.The Mapping Cylinder and Mapping Cone
4.The Hopf Invariant
References
APPENDIX A A Proof of De Rham's Theorem
1.Introduction
2.Differentiable Singular Chains
3.Statement and Proof of De Rham's Theorem
References
APPENDIX B Permutation Groups or Transformation Groups
1.Basic Definitions
2.Homogeneous G-spaces
Index
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