Basic Concept of Algebraic Topology - ISBN:0387902880

Basic Concept of Algebraic Topology

Date: 21 April 2011, 14:04

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Table of contents:
List of Symbols
1. Geometric Complexes and Polyhedra
1.1 Introduction
1.2 Examples
1.2.1 The Jordan Curve Theorem and Related Problems
1.2.2 Integration on Surfaces and Multiply-connected Domains
1.2.3 Classification of Surfaces and Polyhedra
1.3 Geometric Complexes and Polyhedra
1.4 Orientation of Geometric Complexes
EXERCISES
2. Simplicial Homology Groups
2.1 Chains, Cycles, Boundaries, and Homology Groups
2.2 Examples of Homology Groups
2.3 The Structure of Homology Groups
2.4 The Euler-Poincare Theorem
2.5 Pseudomanifolds and the Homology Groups of S^n
EXERCISES
3. Simplicial Approximation
3.1 Introduction
3.2 Simplicial Approximation
3.3 Induced Homomorphisms on the Homology Groups
3.4 The Brouwer Fixed Point Theorem and Related Results
EXERCISES
4. The Fundamental Group
4.1 Introduction
4.2 Homotopic Paths and the Fundamental Group
4.3 The Covering Homotopy Property for S^1
4.4 Examples of Fundamental Groups
4.5 The Relation between H_1(K) and p_1(|.|)
EXERCISES
5. Covering Spaces
5.1 The Definition and Some Examples
5.2 Basic Properties of Covering Spaces
5.3 Classification of Covering Spaces
5.4 Universal Covering Spaces
5.5 Applications
EXERCISES
6. The Higher Homotopy Groups
6.1 Introduction
6.2 Equivalent Definitions of p_n(X, x_0)
6.3 Basic Properties and Examples
6.4 Homotopy Equivalence
6.5 Homotopy Groups of Spheres
EXERCISES
7. Further Developments in Homology
7.1 Chain Derivation
7.2 The Lefschetz Fixed Point Theorem
7.3 Relative Homology Groups
7.4 Singular Homology Theory
7.5 Axioms for Homology Theory
EXERCISES
A Note About the Appendices
APPENDIX 1. Set Theory
APPENDIX 2. Point-set Topology
APPENDIX 3. Algebra
Bibliography
Index

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