دانلود کتاب Algebraic Topology

فهرست مطالب :

Foreword
Introduction
Contents
1 Surface Preliminaries
1.1 Surfaces
1.2 Euclidean Space
1.3 Open Sets
1.4 Functions and Their Properties
1.5 Continuity
1.6 Problems
2 Surfaces
2.1 The Definition of a Surface
2.2 Examples of Surfaces
2.3 Spheres as Surfaces
2.4 Surfaces with Boundary
2.5 Closed, Bounded, and Compact Surfaces
2.6 Equivalence Relations and Topological Equivalence
2.7 Homeomorphic Spaces
2.8 Invariants
2.9 Problems
3 The Euler Characteristic and Identification Spaces
3.1 Triangulations and the Euler Characteristic
3.2 Invariance of the Euler Characteristic
3.3 Identification Spaces
3.4 ID Spaces as Surfaces
3.5 Abstract Topological Spaces
3.6 The Quotient Topology
3.7 Further Examples of ID Spaces
3.8 Triangulations of ID Spaces
3.9 The Connected Sum
3.10 The Euler Characteristic of a Compact Surface with Boundary
3.11 Problems
4 Classification Theorem of Compact Surfaces
4.1 The Geometry of the Projective Plane and the Klein Bottle
4.2 Orientable and Nonorientable Surfaces
4.3 The Classification Theorem for Compact Surfaces
4.4 Compact Surfaces Have Finite Triangulations
4.5 Proof of the Classification Theorem
4.6 Problems
5 Introduction to Group Theory
5.1 Why Use Groups?
5.2 A Motivating Example
5.3 Definition of a Group
5.4 Examples of Groups
5.5 Free Groups, Generators, and Relations
5.6 Free Products
5.7 Problems
6 Structure of Groups
6.1 Subgroups
6.2 Direct Products of Groups
6.3 Homomorphisms
6.4 Isomorphisms
6.5 Existence of Homomorphisms
6.6 Finitely Generated Abelian Groups
6.7 Problems
7 Cosets, Normal Subgroups, and Quotient Groups
7.1 Cosets
7.2 Lagrange\'s Theorem and Its Consequences
7.3 Coset Spaces and Quotient Groups
7.4 Properties and Examples of Normal Subgroups
7.5 Coset Representatives
7.6 A Quotient of a Dihedral Group
7.7 Building up Finite Groups
7.8 An Isomorphism Theorem
7.9 Problems
8 The Fundamental Group
8.1 Paths and Loops on a Surface
8.2 Equivalence of Paths and Loops
8.3 Equivalence Classes of Paths and Loops
8.4 Multiplication of Path and Loop Classes
8.5 Definition of the Fundamental Group
8.6 Problems
9 Computing the Fundamental Group
9.1 Homotopies of Maps and Spaces
9.2 Computing the Fundamental Group of a Circle
9.3 Problems
10 Tools for Fundamental Groups
10.1 More Fundamental Groups
10.2 The Degree of a Loop
10.3 Fundamental Group of a Circle—Redux
10.4 The Induced Homomorphism on Fundamental Groups
10.5 Retracts
10.6 Problems
11 Applications of Fundamental Groups
11.1 The Fundamental Theorem of Algebra
11.2 Further Applications of the Fundamental Group
11.3 Problems
12 The Seifert–Van Kampen Theorem
12.1 Wedges of circles
12.2 The Seifert–Van Kampen Theorem: First Version
12.3 More Fundamental Groups
12.4 The Seifert–Van Kampen Theorem: Second Version
12.5 The Fundamental Group of a Compact Surface
12.6 Even More Fundamental Groups
12.7 Proof of the Second Version of the Seifert–Van Kampen Theorem
12.8 General Seifert–Van Kampen Theorem
12.9 Groups as Fundamental Groups
12.10 Problems
13 Introduction to Homology
13.1 The Idea of Homology
13.2 Chains
13.3 The Boundary Map
13.4 Homology
13.5 The Zeroth Homology Group
13.6 Homology of the Klein Bottle
13.7 Homology and Euler Characteristic
13.8 Homology and Orientability
13.9 Smith Normal Form
13.10 The Induced Map on Homology
13.11 Problems
14 The Mayer–Vietoris Sequence
14.1 Exact Sequences
14.2 The Mayer–Vietoris Sequence
14.3 Homology of Orientable Surfaces
14.4 The Jordan Curve Theorem
14.5 The Hurewicz Map
14.6 Problems
Appendix A Topological Notions
A.1 Compactness Results
A.2 Technical Conditions for Abstract Surfaces
Appendix B A Brief Look at Singular Homology
Appendix C Hints for Selected Problems
Appendix References
Index