An Introduction to Algebraic Topology
by Wallace, Andrew H.Format: Paperback
Pub. Date: 2007-02-27
Publisher(s): Dover Publications
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Summary
This self-contained treatment assumes only some knowledge of real numbers and real analysis. The first three chapters focus on the basics of point-set topology, after which the text proceeds to homology groups and continuous mapping, barycentric subdivision, and simplicial complexes. Exercises form an integral part of the text. 1961 edition.
Author Biography
Andrew H. Wallace, Professor Emeritus at the University of Pennsylvania
Table of Contents
| Prerequisites | |
| Set theory | p. 1 |
| Algebra | p. 3 |
| Euclidean spaces | p. 6 |
| Introduction | |
| Continuity and neighbourhoods | p. 7 |
| The abstract concept of neighbourhood | p. 8 |
| Topological Spaces | |
| Definition of a topological space | p. 14 |
| Open sets | p. 17 |
| Another definition of a topological space | p. 21 |
| Subspaces of a given space | p. 26 |
| Limits | p. 29 |
| Limit points | p. 33 |
| Closure of a set | p. 36 |
| Frontier of a set | p. 40 |
| Topological Properties of Spaces | |
| Continuous mappings and homeomorphisms | p. 42 |
| Compact spaces | p. 50 |
| Arcwise connected spaces | p. 55 |
| Connected spaces | p. 57 |
| The Fundamental Group | |
| Homotopy | p. 63 |
| Homotopy classes | p. 69 |
| The fundamental group | p. 74 |
| Change of base-point | p. 85 |
| Topological invariance | p. 89 |
| The Homology Groups | |
| Geometrical motivation for homology theory | p. 92 |
| Euclidean simplexes | p. 95 |
| Linear mappings | p. 100 |
| Singular simplexes on a space | p. 103 |
| Chains on a space | p. 104 |
| The boundary of a simplex | p. 105 |
| Boundaries and cycles on any space | p. 107 |
| Homologous cycles and homology groups | p. 111 |
| Relative homology | p. 113 |
| Continuous Mappings and the Homology Groups | |
| The induced homomorphism | p. 117 |
| Topological invariance of the homology groups | p. 119 |
| Homotopic mappings and the homology groups | p. 120 |
| Prisms | p. 123 |
| Homotopic mappings and the homology groups (contd.) | p. 128 |
| Barycentric Subdivision and Excision | |
| Motivation for barycentric subdivision | p. 133 |
| The operator B | p. 137 |
| The operator H | p. 140 |
| Reduction to small simplexes | p. 142 |
| The excision theorem | p. 146 |
| The Homology Sequence | |
| The exact sequence | p. 150 |
| Homology groups in some special cases | p. 153 |
| Homology groups in cells and spheres | p. 162 |
| Simplicial Complexes | |
| Definition of complexes | p. 168 |
| The direct sum theorem for complexes | p. 171 |
| A generator for H[subscript r](S,S) | p. 175 |
| The homology groups of a simplicial complex | p. 180 |
| Oriented chains and cycles | p. 184 |
| The oriented boundary operator | p. 188 |
| Guide to Further Reading | p. 195 |
| Index | p. 197 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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