Contents Chapter 1.Basic Concepts 4 1. Terminology and Notations 4 1.1. Set Theory 4 1.2. Logical Equivalence 4 1.3. Topological Spaces 5 1.4. Operations on Topological Spaces 5 1.5. Operations on Pointed Spaces 8 2. Homotopy 10 2.1. Homotopies 10 2.2. Paths 10 2.3. Homotopy as a Path 11 2.4. Homotopy Equivalence 11 2.5. Retractions 11 2.6. Deformation Retractions 12 2.7. Relative Homotopies 13 2.8. k-connectedness 13 2.9. Borsuk Pairs 14 2.10. CNRS Spaces 15 2.11. Homotopy Properties of Topological Constructions 15 2.12. Natural Group Structures on Sets of Homotopy Classes 16 3. Homotopy Groups 20 3.1. Absolute Homotopy Groups 20 3.2. Digression： Local Systems 22 3.3. Local Systems of Homotopy Groups of a Topological Space 23 3.4. Relative Homotopy Groups 25 3.5. The Homotopy Sequence of a Pair 28 3.6. Splitting 31 3.7. The Homotopy Sequence of a Triple 32 Chapter 2.Bundle Techniques 33 4. Bundles 33 4.1. General Definitions 33 4.2. Locally Trivial Bundles 34 4.3. Serre Bundles 36 4.4. Bundles of Spaces of Maps 37 5. Bundles and Homotopy Groups 38 5.1. The Local System of Homotopy Groups of the Fibres of a Serre Bundle 38 5.2. The Homotopy Sequence of a Serre Bundle 39 5.3. Important Special Cases 40 6. The Theory of Coverings 41 6.1. Coverings 41 6.2. The Group of a Covering 42 6.3. Hierarchies of Coverings 42 6.4. The Existence of Coverings 43 6.5. Automorphisms of a Coveting 44 6.6. Regular Coverings 44 6.7. Covering Maps 45 Chapter 3 Cellular Techniques 45 7. Cellular Spaces 45 7.1. Basic Concepts 45 7.2. Gluing of Cellular Spaces from Balls 48 7.3. Examples of Cellular Decompositions 49 7.4. Topological Properties of Cellular Spaces 52 7.5. Cellular Constructions 53 8. Simplicial Spaces 54 8.1. Basic Concepts 54 8.2. Simplicial Schemes 58 8.3. Simplicial Constructions 59 8.4. Stars, Links, Regular Neighbourhoods 62 8.5. Simplicial Approximation of a Continuous Map 64 9. Cellular Approximation of Maps and Spaces 64 9.1. Cellular Approximation of a Continuous Map 64 9.2. Cellular k-connected Pairs 65 9.3. Simplicial Approximation of Cellular Spaces 66 9.4. Weak Homotopy Equivalence 67 9.5. Cellular Approximation to Topological Spaces 69 9.6. The Covering Homotopy Theorem 71 Chapter 4 The Simplest Calculations 72 10. The Homotopy Groups of Spheres and Classical Manifolds 72 10.1. Suspension in the Homotopy Groups of Spheres 72 10.2. The Simplest Homotopy Groups of Spheres 73 10.3. The Composition Product 74 10.4. Homotopy Groups of Spheres 75 10.5. Homotopy Groups of Projective Spaces and Lens Spaces 77 10.6. Homotopy Groups of the Classical Groups 78 10.7. Homotopy Groups of Stiefel Manifolds and Spaces 79 10.8. Homotopy Groups of Grassmann Manifolds and Spaces 80 11. Application of Cellular Techniques 81 11.1. Homotopy Groups of a 1-dimensional Cellular Space 81 11.2. The Effect of Attaching Balls 81 11.3. The Fundamental Group of a Cellular Space 83 11.4. Homotopy Groups of Compact Surfaces 84 11.5. Homotopy Groups of Bouquets 85 11.6. Homotopy Groups of a k-connected Cellular Pair 86 11.7. Spaces with Given Homotopy Groups 87 12. Appendix 89 12.1. The Whitehead Product 89 12.2. The Homotopy Sequence of a Triad 91 12.3. Homotopy Excision, Quotient and Suspension Theorems 93
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