Topology is a branch of mathematics packed with intriguing concepts, fascinating geometrical objects, and ingenious methods for studying them. The authors have written this textbook to make this material accessible to undergraduate students who may be at the beginning of their study of upper-level mathematics and who may not have covered the extensive prerequisites required for a traditional course in topology. The approach is to cultivate the intuitive ideas of continuity, convergence, and connectedness so students can quickly delve into knot theory, the topology of surfaces, and three-dimensional manifolds, fixed points, and elementary homotopy theory. The fundamental concepts of point-set topology appear at the end of the book when students can see how this level of abstraction provides a sound logical basis for the geometrical ideas that have come before. This organization presents students with the exciting geometrical ideas of topology now(!) rather than later.
Anyone using this book should have some exposure to the geometry of objects in higher-dimensional Euclidean spaces together with an appreciation of precise mathematical definitions and proofs. Multivariable calculus, linear algebra, and one further proof-oriented mathematics courses are suitable preparation.
Author(s): Robert Messer, Philip Straffin
Series: Classroom Resource Material
Edition: 1
Publisher: The Mathematical Association of America
Year: 2006
Language: English
Pages: 250
Front Cover......Page 1
Contents......Page 3
Preface......Page 5
1.1 Equivalence......Page 9
1.2 Bijections......Page 14
1.3 Continuous Functions......Page 22
1.4 Topological Equivalence......Page 28
1.5 Topological Invariants......Page 33
1.6 Isotopy......Page 40
References & Suggested Readings for Ch. 1......Page 48
2.1 Knots, Links, & Equivalences......Page 49
2.2 Knot Diagrams......Page 55
2.3 Reidemeister Moves......Page 63
2.4 Colorings......Page 69
2.5 The Alexander Polynomial......Page 73
2.6 Skein Relations......Page 86
2.7 The Jones Polynomial......Page 90
References & Suggested Readings for Ch. 2......Page 96
3.1 Definitions & Examples......Page 99
3.2 Cut-and-Paste Techniques......Page 105
3.3 The Euler Characteristic & Orientability......Page 111
3.4 Classification of Surfaces......Page 117
3.5 Surfaces Bounded by Knots......Page 128
References & Suggested Readings for Ch. 3......Page 133
4.1 Definitions & Examples......Page 135
4.2 Euler Characteristic......Page 139
4.3 Gluing Polyhedral Solids......Page 143
4.4 Heegaard Splittings......Page 151
References & Suggested Readings for Ch. 4......Page 158
5.1 Continuous Functions on Closed Bounded Intervals......Page 159
5.2 Contraction Mapping Theorem......Page 164
5.3 Spemer's Lemma......Page 168
5.4 Brouwer Fixed-Point Theorem for a Disk......Page 171
References & Suggested Readings for Ch. 5......Page 175
6.1 Deformations with Singularities......Page 177
6.2 Algebraic Properties......Page 182
6.3 Invariance of the Fundamental Group......Page 187
6.4 The Sphere & the Circle......Page 192
6.5 Words & Relations......Page 200
6.6 The Poincare Conjecture......Page 209
References & Suggested Readings for Ch. 6......Page 216
7.1 Metric Spaces......Page 217
7.2 Topological Spaces......Page 225
7.3 Connectedness......Page 230
7.4 Compactness......Page 234
7.5 Quotient Spaces......Page 237
References & Suggested Readings for Ch. 7......Page 239
Index......Page 241
About the Authors......Page 247
Back Cover......Page 250
