Essential Topology

Description

Taking a direct route, "Essential Topology" brings the most important aspects of modern topology within reach of a second-year undergraduate student. It begins with a discussion of continuity and, by way of many examples, leads to the celebrated "Hairy Ball theorem" and on to homotopy and homology: the cornerstones of contemporary algebraic topology. While containing all the key results of basic topology, Essential Topology never allows itself to get mired in details. Instead, the focus throughout is on providing interesting examples that clarify the ideas and motivate the student. With chapters on: continuity and topological spaces; deconstructionist topology; the Euler number; homotopy groups including the fundamental group; simplicial and singular homology, and fibre bundles. "Essential Topology" contains enough material for two semester-long courses, and offers a one-stop-shop for undergraduate-level topology, leaving students motivated for postgraduate study in the field, and well-prepared for it. Preface Contents 1. Introduction 2. Continuous Functions 2.1 Naïve Continuity 2.2 Rigorous Continuity 2.3 Open Sets 2.4 Continuity by Open Sets 3. Topological Spaces 3.1 Topological Spaces 3.2 More Examples of Topological Spaces 3.3 Continuity in the Subspace Topology 3.4 Bases Interlude 4. Topological Properties 4.1 Connectivity 4.2 Compactness 4.3 The Hausdorff Property 5. Deconstructionist Topology 5.1 Homeomorphisms 5.2 Disjoint Unions 5.3 Product Spaces 5.4 Quotient Spaces Interlude 6. Homotopy 6.1 Homotopy 6.2 Homotopy Equivalence 6.3 The Circle 6.4 Brouwer's Fixed-Point Theorem 6.5 Vector Fields 7. The Euler Number 7.1 Simplicial Complexes 7.2 The Euler Number 7.3 The Euler Characteristic and Surfaces 8. Homotopy Groups 8.1 Homotopy Groups 8.2 Induced Homomorphisms 8.3 The Fundamental Group 8.4 Path Connectivity and 0 8.5 The Van Kampen Theorem 9. Simplicial Homology 9.1 Simplicial Homology Modulo 2 9.2 Limitations of Homology Modulo 2 9.3 Integral Simplicial Homology 10. Singular Homology 10.1 Singular Homology 10.2 Homology and Continuous Maps 10.3 Homology Respects Homotopies 10.4 Barycentric Subdivision 10.5 The Mayer--Vietoris Sequence 10.6 Homology and Homotopy Groups 10.7 Comparison of Singular and Simplicial Homology 11. More Deconstructionism 11.1 Wedge Products 11.2 Suspensions and Loop Spaces 11.3 Fibre Bundles 11.4 Vector Bundles Solutions to Selected Exercises Bibliography Index