This textbook provides a succinct introduction to algebraic topology. It follows a modern categorical approach from the beginning and gives ample motivation throughout so that students will find this an ideal first encounter to the field. Topics are treated in a self-contained manner, making this a convenient resource for instructors searching for a comprehensive overview of the area. It begins with an outline of category theory, establishing the concepts of functors, natural transformations, adjunction, limits, and colimits. As a first application, van Kampen's theorem is proven in the groupoid version. Following this, an excursion to cofibrations and homotopy pushouts yields an alternative formulation of the theorem that puts the computation of fundamental groups of attaching spaces on firm ground. Simplicial homology is then defined, motivating the Eilenberg-Steenrod axioms, and the simplicial approximation theorem is proven. After verifying the axioms for singular homology, various versions of the Mayer-Vietoris sequence are derived and it is shown that homotopy classes of self-maps of spheres are classified by degree.The final chapter discusses cellular homology of CW complexes, culminating in the uniqueness theorem for ordinary homology. Introduction to Algebraic Topology is suitable for a single-semester graduate course on algebraic topology. It can also be used for self-study, with numerous examples, exercises, and motivating remarks included. Preface 6 Contents 8 1 Basic Notions of Category Theory 10 1.1 Categories 10 1.2 Functors 13 1.3 Natural Transformations 15 1.4 Adjunction 18 1.5 Limits and Colimits 20 Exercises 38 2 Fundamental Groupoid and van Kampen's Theorem 41 2.1 The Fundamental Groupoid 41 2.2 Van Kampen's Theorem 43 2.3 Cofibrations and Homotopy Pushouts 49 2.4 Computing Fundamental Groups 61 2.5 Higher Homotopy Groups 64 Exercises 66 3 Homology: Ideas and Axioms 67 3.1 The Idea of Homology 67 3.2 Simplicial Homology 69 3.3 Relative Simplicial Homology with Coefficients 74 3.4 The Eilenberg–Steenrod Axioms for Homology 79 3.5 Simplicial Approximation 81 Exercises 85 4 Singular Homology 87 4.1 The Definition of Singular Homology 87 4.2 The Long Exact Sequence of a Pair of Spaces 89 4.3 Homotopy Invariance 92 4.4 Excision 96 4.5 Singular Homology in Degree Zero and One 104 Exercises 108 5 Homology: Computations and Applications 110 5.1 Relative vs. Absolute Homology 110 5.2 Simplicial and Singular Homology Agree 117 5.3 The Mayer–Vietoris Sequence 121 5.4 Degree 127 5.5 Applications 133 The Fundamental Theorem of Algebra 133 Invariance of Dimension 134 Nonexistence of Retractions 134 The Brouwer Fixed Point Theorem 135 The Borsuk–Ulam Theorem 135 The Ham Sandwich Theorem 138 Exercises 139 6 Cellular Homology 141 6.1 CW Complexes 141 6.2 Cellular Homology and Euler Characteristic 152 6.3 Computing Cellular Homology 161 6.4 Uniqueness of Ordinary Homology 166 6.5 How to Proceed 170 Exercises 173 A Quotient Topology 174 Bibliography 177 List of Notation 174 List of Notation 179 Index 182 Basic notions of category theory -- Fundamental groupoid and van Kampen's theorem -- Homology: ideas and axioms -- Singular homology -- Homology: computations and applications -- Cellular homology -- Appendix: Quotient topology
