Algebraic topology: A first course [1 ed.] 0824763513, 9780824763510

Table of contents :
Front Cover
Contents
Preface
1: SOME OLD TOPOLOGICAL PROBLEMS
1.0. Introduction
1.1. The Descartes-Euler Theorem
1.2. Coloring Maps and Graphs
1.3. The Jordan Curve Theorem; Knots
1.4. Further Early Investigations
2: THE GEOMETRY OF COMPLEXES
2.1. What is Topology
2.2. Simplicial Complexes and Maps; Polyhedra
2.3. Abstract Simplicial Complexes; Cutting and Pasting
2.4. Historical Comments
3: THE CLASSIFICATION OF SURFACES
3.0. Introduction
3.1. The Definition of a Surface
3.2. Representing Surfaces by Symbols
3.3. The Normal Form for Some Surfaces
3.4. The Classification Theorems
3.5. Bordered and Noncompact Surfaces
3.6. Historical Comments
4: THE HOMOLOGY GROUPS
4.1. Some Motivation
4.2. The Orientation of a Simplex
4.3. The Definition of the Homology Groups
4.4. The Homology Groups of a Cone
4.5. The Topological Invariance of Homology Groups
4.6. Historical Comments
5: MAPS AND HOMOTOPY
5.1. Simplicial Maps Again
5.2. Homotopy
5.3. Simplicial Approximations
5.4. The Barycentric Subdivision
5.5. The Simplicial Approximation Theorem; Induced Homomorphisms
5.6. Historical Comments
6: FIRST APPLICATIONS OF HOMOLOGY THEORY
6.1. A Quick Review
6.2. Local Homology Groups
6.3. Some Invariance Theorems
6.4. Homology with Arbitrary Coefficients; The Mod 2 Homology Groups
6.5. Pseudomanifolds; Orientability
6.6. Euler's Theorem Revisited
6.7. Retracts and the Brouwer Fixed-Point Theorem
6.8. Historical Comment
7: MAPS OF SPHERES AND MORE APPLICATIONS
7.1. The Degree of a Map; Vector Fields
7.2. The Borsuk-Ulam Theorem and the Ham Sandwich Problem
7.3. More on Degrees of Maps; Zeros of Polynomials More on Degrees of Maps; Zeros of Polynomials
7.4. Local Degrees, Solvability of Equations and Some Complex Analysis
7.5. Extending Maps
7.6. The Jordan Curve Theorem and Other Separation Theorems
7.7. Historical Comments
8: CONCLUDING REMARKS
APPENDIX A: The Topology of R^n
APPENDIX B: Permutations and Abelian Groups
APPENDIX C: The Incidence Matrices
List of Symbols
Bibliography
Index

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