Introduction to Modern Topology and Geometry Anatole Katok Alexey Sossinsky Contents Chapter 1. BASIC TOPOLOGY 3 1.1. Topological spaces 3 1.2. Continuous maps and homeomorphisms 6 1.3. Basic constructions 9 1.4. Separation properties 14 1.5. Compactness 19 1.6. Connectedness and path connectedness 22 1.7. Totally disconnected spaces and Cantor sets 26 1.8. Topological manifolds 28 1.9. Orbit spaces for group actions 32 1.10. Problems 35 Chapter 2. ELEMENTARY HOMOTOPY THEORY 39 2.1. Homotopy and homotopy equivalence 39 2.2. Contractible spaces 42 2.3. Graphs 44 2.4. Degree of circle maps 46 2.5. Brouwer fixed point theorem in dimension two 50 2.6. Index of a point w.r.t. a curve 52 2.7. The fundamental theorem of algebra 53 2.8. The fundamental group; definition and elementary properties 55 2.9. The first glance at covering spaces 62 2.10. Definition of higher homotopy groups 65 2.11. Hopf fibration 67 2.12. Problems 69 Chapter 3. METRICS AND RELATED STRUCTURES 71 3.1. Definition of metric spaces and basic constructions 71 3.2. Cauchy sequences and completeness 75 3.3. The p-adic completion of integers and rationals 78 3.4. Maps between metric spaces 80 3.5. Role of metrics in geometry and topology 84 3.6. Separation properties and metrizability 85 3.7. Compact metric spaces 86 3.8. Metric spaces with symmetries and self-similarities 90 3.9. Spaces of continuous maps 92 3.10. Spaces of closed subsets of a compact metric space 93 1 2 CONTENTS 3.11. Uniform structures 95 3.12. Topological groups 96 3.13. Problems 96 Chapter 4. REAL AND COMPLEX SMOOTH MANIFOLDS 99 4.1. Differentiable manifolds, smooth maps and diffeomorphisms 99 4.2. Principal constructions 103 4.3. Orientability and degree 105 4.4. Paracompactness and partition of unity 105 4.5. Embedding into Euclidean space 107 4.6. Derivatives and the tangent bundle 108 4.7. Smooth maps and the tangent bundle 112 4.8. Manifolds with boundary 114 4.9. Complex manifolds 115 4.10. Lie groups: first examples 117 4.11. Problems 120 Chapter 5. TOPOLOGY AND GEOMETRY OF SURFACES 123 5.1. Two big separation theorems: Jordan and Schoenflies 123 5.2. Planar and non-planar graphs 125 5.3. Surfaces and their triangulations 127 5.4. Euler characteristic and genus 130 5.5. Classification of surfaces 132 5.6. The fundamental group of compact surfaces 135 5.7. Vector fields on the plane 136 5.8. Smoothing surfaces 139 5.9. Vector fields on surfaces 139 5.10. The geometry of Riemannian metrics 143 5.11. Gauss–Bonnet theorem 143 5.12. Complex structure on surfaces 144 Chapter 6. COVERING SPACES AND DISCRETE GROUPS 145 6.1. Coverings associated with discrete group actions 145 6.2. The hierarchy of coverings, universal coverings 149 6.3. Path lifting and covering homotopy properties 151 6.4. Classification of coverings with given base via π1 153 6.5. Coverings of surfaces and the Euler characteristic 159 6.6. Branched coverings of surfaces 161 6.7. Riemann–Hurwitz formula 164 6.8. Problems 169 Chapter 7. SIMPLICIAL AND CW SPACES 171 7.1. Simplicial spaces and maps 171 7.2. Simplicial approximation 181 7.3. Triangulating manifolds 185 7.4. CW-spaces, cellular maps 190.