Essential topology

Description

This book brings the most important aspects of modern topology within reach of a second-year undergraduate student. It successfully unites the most exciting aspects of modern topology with those that are most useful for research, leaving readers prepared and motivated for further study. Written from a thoroughly modern perspective, every topic is introduced with an explanation of why it is being studied, and a huge number of examples provide further motivation. The book is ideal for self-study and assumes only a familiarity with the notion of continuity and basic algebra. front-matter.pdf......Page 1 Preface......Page 5 Contents......Page 7 1. Introduction......Page 10 2.1 Naïve Continuity......Page 12 2.2 Rigorous Continuity......Page 14 2.3 Open Sets......Page 17 2.4 Continuity by Open Sets......Page 20 3.1 Topological Spaces......Page 24 3.2 More Examples of Topological Spaces......Page 28 3.3 Continuity in the Subspace Topology......Page 37 3.4 Bases......Page 40 4.1 Connectivity......Page 44 4.2 Compactness......Page 50 4.3 The Hausdorff Property......Page 57 5.1 Homeomorphisms......Page 61 5.2 Disjoint Unions......Page 72 5.3 Product Spaces......Page 77 5.4 Quotient Spaces......Page 82 6.1 Homotopy......Page 95 6.2 Homotopy Equivalence......Page 100 6.3 The Circle......Page 106 6.4 Brouwer's Fixed-Point Theorem......Page 114 6.5 Vector Fields......Page 116 7.1 Simplicial Complexes......Page 121 7.2 The Euler Number......Page 124 7.3 The Euler Characteristic and Surfaces......Page 127 8. Homotopy Groups......Page 131 8.1 Homotopy Groups......Page 132 8.2 Induced Homomorphisms......Page 140 8.3 The Fundamental Group......Page 144 8.4 Path Connectivity and 0......Page 145 8.5 The Van Kampen Theorem......Page 148 9. Simplicial Homology......Page 153 9.1 Simplicial Homology Modulo 2......Page 154 9.2 Limitations of Homology Modulo 2......Page 162 9.3 Integral Simplicial Homology......Page 164 10.1 Singular Homology......Page 171 10.2 Homology and Continuous Maps......Page 177 10.3 Homology Respects Homotopies......Page 179 10.4 Barycentric Subdivision......Page 184 10.5 The Mayer--Vietoris Sequence......Page 190 10.6 Homology and Homotopy Groups......Page 198 10.7 Comparison of Singular and Simplicial Homology......Page 199 11.1 Wedge Products......Page 202 11.2 Suspensions and Loop Spaces......Page 204 11.3 Fibre Bundles......Page 209 11.4 Vector Bundles......Page 214 Solutions to Selected Exercises......Page 218 Bibliography......Page 222 Index......Page 224