Topology II: Homotopy and Homology. Classical Manifoldsto Homotopy Theory O. Ya. Viro, D. B. Fuchs Translated from the Russian by C. J. Shaddock 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 2 O. Ya. Viro, D. B. Fuchs 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . |
Contents
4 | |
5 | |
10 | |
5 | 28 |
Bundle Techniques | 33 |
8 | 40 |
6 The Theory of Coverings | 41 |
8 Simplicial Spaces | 59 |
11 Application of Cellular Techniques | 81 |
38 | 87 |
12 Appendix | 89 |
Homology and Cohomology | 94 |
3 Homotopy Groups | 125 |
41 | 209 |
42 | 216 |
251 | |
Other editions - View all
Topology II: Homotopy and Homology. Classical Manifolds D.B. Fuchs,O.Ya. Viro No preview available - 2013 |
Common terms and phrases
Abelian group algebraic axiom barycentric base point xo Borsuk pair boundary c₁ called canonical cell decomposition cellular approximation cellular decomposition cellular pair cellular space characteristic class cochain coefficients compact complex construction continuous map corresponding covering Cq(X defined denoted diffeomorphic dimension elements embedding epimorphism Euclidean Euler exact sequence example fibre finite cellular fixed points follows formula fundamental group Hence homology and cohomology homology theory homomorphism homotopy class homotopy groups homotopy sequence Hopf Hq(C Hq(X inclusion induced integer intersection number inverse isomorphism k-connected lemma map f map ƒ n-dimensional oriented particular polynomials Pontryagin classes projection proof quaternionic retract Serre bundle simplicial space singular simplex smooth manifold SO(n sphere spheroid Steenrod Stiefel-Whitney classes subgroup submanifold subset subspace theorem topological space triangulation vector bundle weak homotopy equivalence Y₁ Young diagram