Two top experts in topology, O.Ya. Viro and D.B. Fuchs, give an up-to-date account of research in central areas of topology and the theory of Lie groups. They cover homotopy, homology and cohomology as well as the theory of manifolds, Lie groups, Grassmannians and low-dimensional manifolds. Their book will be used by graduate students and researchers in mathematics and mathematical physics.
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目录
I. Introduction to Homotopy Theory Chapter 1 Basic Concepts §1. Terminology and Notations 1.1. Set Theory 1.2. Logical Equivalence 1.3. Topological Spaces 1.4. Operations on Topological Spaces 1.5. Operations on Pointed Spaces §2. Homotopy 2.1. Homotopies 2.2. Paths 2.3. Homotopy as a Path 2.4. HomotopyEquivalence 2.5. Retractions 2.6. Deformation Retractions 2.7. Relative Homotopies 2.8. k-connectedness 2.9. Borsuk Pairs 2.10. CNRS Spaces 2.11. Homotopy Properties of Topological Constructions. 2.12. Natural Group Structures on Sets of Homotopy Classes §3. Homotopy Groups 3.1. Absolute Homotopy Groups 3.2. Digression: Local Systems 3.3. Local Systems of Homotopy Groups of a Topological Space 3.4. Relative Homotopy Groups 3.5. The Homotopy Sequence of a Pair 3.6. Splitting 3.7. The Homotopy Sequence of a Triple Chapter 2. Bundle Techniques §4. Bundles 4.1. General Definitions 4.2. Locally Trivial Bundles 4.3. Serre Bundles 4.4. Bundles of Spaces of Maps §5. Bundles and Homotopy Groups 5.1. The Local System of Homotopy Groups of the Fibres of a Serre Bundle 5.2. The Homotopy Sequence of a Serre Bundle 5.3. Important Special Cases §6. The Theory of Coverings 6.1. Coverings 6.2. The Group of a Covering 6.3. Hierarchies of Coverings 6.4. The Existence of Coverings 6.5. Automorphisms of a Covering 6.6. Regular Coverings 6.7. Covering Maps Chapter 3 Cellular Techniques §7. Cellular Spaces 7.1. Basic Concepts 7.2. Gluing of Cellular Spaces from Balls 7.3. Examples of Cellular Decompositions 7.4. Topological Properties of Cellular Spaces 7.5. Cellular Constructions §8. Simplicia1 Spaces 8.1. Basic Concepts 8.2. Simplicia1 Schemes 8.3. Simplicia1 Constructions 8.4. Stars, Links, Regular Neighbourhoods 8.5. Simplicia1 Approximation of a Continuous Map §9. Cellular Approximation of Maps and Spaces 9.1. Cellular Approximation of a Continuous Map 9.2. Cellular k-connected Pairs 9.3. Simplicia1 Approximation of Cellular Spaces 9.4. Weak Homotopy Equivalence 9.5. Cellular Approximation to Topological Spaces 9.6. The Covering Homotopy Theorem Chapter 4 The Simplest Calculations §10. The Homotopy Groups of Spheres and Classical Manifolds 10.1. Suspension in the Homotopy Groups of Spheres 10.2. The Simplest Homotopy Groups of Spheres 10.3. The Composition Product 10.4. Homotopy Groups of Spheres 10.5. Homotopy Groups of Projective Spaces and Lens Spaces 10.6. Homotopy Groups of the Classical Groups 10.7. Homotopy Groups of Stiefel Manifolds and Spaces 10.8. Homotopy Groups of Grassmann Manifolds and Spaces §11. Application of Cellular Techniques 11.1. Homotopy Groups of a 1-dimensional Cellular Space 11.2. The Effect of Attaching Balls 11.3. The Fundamental Group of a Cellular Space 11.4. Homotopy Groups of Compact Surfaces 11.5. Homotopy Groups of Bouquets 11.6. Homotopy Groups of a k-connected Cellular Pair 11.7. Spaces with Given Homotopy Groups §12. Appendix 12.1. The Whitehead Product 12.2. The Homotopy Sequence of a Triad 12.3. Homotopy Excision, Quotient and Suspension Theorems II. Homology and Cohomology Chapter 1 Additive Theory §1. Algebraic Preparation 1.1. Complexes and Their Homology 1.2. Maps and Homotopies 1.3. Homology sequences 1.4. The Euler characteristic and the Lefschetz number 1.5. Change of coefficients 1.6. Tensor products of complexes and the Kiinneth formula §2. General singular homology theory 2.1. Basic definitions 2.2. The simplest calculations 2.3. Natural transformations；refinement and approximation 2.4. Excision，factorization，suspension 2.5. Addition theorems 2.6. Dependence on the coefficients §3. Homology of cellular spaces 3.1. The cellular complex 3.2. Interrelations with the singular complex 3.3. The simplicia] case 3.4. Examples of calculations 3.5. Other applications §4. Homology and homotopy 4.1. Weak homotopy equivalence and homology 4.2. The Hurewicz theorems 4.3. The theorems of Poincare and Hopf 4.4. Whitehead's theorem 4.5. Some instructive examples §5. Homology and fixed points 5.1. Lefschetz's theorem 5.2. Smith theory §6. Other homology and cohomology theories 6.1. The Eilenberg-Steenrod axioms 6.2. An alternative construction of the Eilenberg-Steemod homology and cohomology theory：the Aleksandrov-Cech theory 6.3. Extraordinary theories 6.4. Homology and cohomology with local coefficients 6.5. Cohomology with coefficients in a sheaf 6.6. Conclusion Chapter 2 Multiplicative theory §7. Products 7.1. Introduction 7.2. Direct construction of the U-product 7.3. Application：the Hopf invariant 7.4. Other products §8. Homology and manifolds 8.1. Introduction 8.2. The fundamental class 8.3. The Poincar6 isomorphisms 8.4. Intersection numbers and Poincart duality 8.5. Linking coefficients 8.6. Inverse homomorphisms 8.7. The relation with the U-product 8.8. Generalizations of the Poincar6 isomorphism and duality Chapter 3 Obstructions. characteristic classes and cohomology operations §9. Obstructions 9.1. Obstructions to extending a continuous map 9.2. The relative case 9.3. Application: cohomology and maps into K(π，n) spaces 9.4. Another application: Hopf s theorems 9.5. Obstructions to the extension of sections §10. Characteristic classes of vector bundles 10.1. Vector bundles 10.2. Associated bundles and characteristic classes 10.3. Characteristic classes and classifying spaces 10.4. The most important properties of Stiefel-Whitney classes 10.5. The most important properties of Euler. Chern. and Pontryagin classes 10.6. Characteristic classes in the topology of smooth manifolds §11. Steenrod squares 11.1. General theory of cohomology operations 11.2. Steenrod squares and their properties 11.3. Steenrod squares and Stiefel-Whitney classes 11.4. Secondary obstructions 11.5. The non-existence of spheroids with odd Hopf invariant References
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