Introductory topology

Stewart S Cairns

Table of contents :
Cover……Page 1
Title page……Page 3
Date-line……Page 4
Dedication……Page 5
Preface……Page 7
Contents……Page 9
Title……Page 13
1-1. The Seven-Bridges Problem……Page 15
1-2. Unicursal Graphs……Page 16
1-3. The Cyclomatic Number……Page 18
1-4. Restrictions on Polyhedra……Page 21
2-1. Polygonal Regions with Matched Edges……Page 27
2-2. Some Elementary Surfaces……Page 32
2-3. Orientability and Non-Orientability……Page 38
2-4. Standard Form for Spheres with Contours and Handles or Crosscaps……Page 41
2-5. A Classification Theorem……Page 45
3-1. Sets and Mappings……Page 53
3-2. Relations, Cartesian Products, Functions……Page 56
3-3. Continuity for Real Functions of Real Variables……Page 60
3-4. Topological Spaces……Page 63
3-5. Homeomorphisms; Definition of Topology……Page 66
3-6. Metric Spaces……Page 68
3-7. Compact Spaces……Page 71
3-8. Brouwer Dimension; The Lebesgue Number……Page 73
4-1. Linear and Convex Subspaces of $E^n$……Page 77
4-2. Dimension Numbers in $E^n$……Page 79
4-3. Barycentric Coordinates……Page 82
4-4. Simplexes……Page 83
4-5. Complexes……Page 86
4-6. Polyhedra; Topological Complexes……Page 87
4-7. Abstract and Generalized Complexes……Page 89
4-8. Realizations of Abstract Complexes……Page 90
4-9. Isomorphisms and Homeomorphisms……Page 92
4-10. Simplicial Mappings……Page 93
4-11. Barycentric Subdivisions……Page 94
4-12. General Polyhedral Complexes……Page 98
5-1. Chains, Cycles, and Bounding Cycles……Page 102
5-2. Homology Groups of Finite Simplicial Complexes……Page 108
5-3. Some Lower-dimensional Cases……Page 110
5-4. Homology Groups of a Surface……Page 111
5-5. Surface Topology……Page 118
5-6. Pseudomanifolds……Page 122
5-7. Homology Bases and Incidence Matrices……Page 125
5-8. Connectivity Groups and Numbers……Page 132
5-9. Cohomology Groups……Page 134
5-10. Dual Bases……Page 137
5-11. Comments on Cohomology Groups……Page 138
6-1. Singular Simplexes……Page 140
6-2. Singular $k$-Chains and Groups……Page 142
6-3. Sperner’s Lemma; Invariance of Dimension……Page 145
6-4. The Brouwer Fixed-Point Theorem……Page 148
6-5. Invariance of Regionality……Page 151
6-6. Singular and Simplicial Groups on a Topological Polyhedron……Page 153
6-7. Simplicial Subsets of Singular Homology Classes……Page 154
6-8. Chains on Prism Complexes……Page 156
6-9. Invariance of Homology Properties……Page 160
6-10. Classes of Mappings……Page 163
7-1. Some Homology Properties of Pseudomanifolds……Page 165
7-2. The $m$-Sphere……Page 166
7-3. Projective $m$-Space……Page 167
7-4. Local Homology Groups……Page 171
7-5. Topological Manifolds and Homology Manifolds……Page 174
7-6. Cell Complexes……Page 176
7-7. Cellular Subdivisions of a Homology Manifold……Page 179
7-8. The Poincare Duality Theorem……Page 183
7-9. Relative Homology……Page 187
7-10. The Lefschetz Duality Theorem……Page 189
7-11. The Alexander Duality Theorem and Consequences……Page 192
8-1. Paths and Path Products……Page 197
8-2. The Fundamental Group……Page 199
8-3. Relation Between $Phi(Sigma)$ and $mathfrak{H}_1(Sigma)$……Page 205
8-4. The Fundamental Groups of $E^n$ and of a Circle……Page 211
8-5. The Fundamental Group of a Surface……Page 213
8-6. Covering Complexes……Page 217
8-7. Fundamental Groups and Coverings……Page 220
Bibliography……Page 223
A-1. Basic Terminology……Page 227
A-2. Homomorphisms and Isomorphisms……Page 231
A-3. The Structure of Finitely Generated Abelian Groups……Page 233
A-4. Integral Modules, Contravariant and Covariant Components……Page 242
A-5. Dual Bases in a Module……Page 245
Index of Symbols……Page 249
General Index……Page 251