I Introductory Notions. - 1. The Fundamental Problems: Extension Homotopy and Classification. - 2. Standard Notations and Conventions. - 3. Maps of the n-sphere into Itself. - 4. Compactly Generated Spaces. - 5. NDR-pairs. - 6. Filtered Spaces. - 7. Fibrations. - II CW-complexes. - 1. Construction of CW-complexes. - 2. Homology Theory of CW-complexes. - 3. Compression Theorems. - 4. Cellular Maps. - 5. Local Calculations. - 6. Regular Cell Complexes. - 7. Products and the Cohomology Ring. - III Generalities on Homotopy Classes of Mappings. - 1. Homotopy and the Fundamental Group. - 2. Spaces with Base Points. - 3. Groups of Homotopy Classes. - 4. H-spaces. - 5. H'-spaces. - 6. Exact Sequences of Mapping Functors. - 7. Homology Properties of H-spaces and H'-spaces. - 8. Hopf Algebras. - IV Homotopy Groups. - 1. Relative Homotopy Groups. - 2. The Homotopy Sequence. - 3. The Operations of the Fundamental Group on the Homotopy Sequence. - 4. The Hurewicz Map. - 5. The Eilenberg and Blakers Homology Groups. - 6. The Homotopy Addition Theorem. - 7. The Hurewicz Theorems. - 8. Homotopy Relations in Fibre Spaces. - 9. Fibrations in Which the Base or Fibre is a Sphere. - 10. Elementary Homotopy Theory of Lie Groups and Their Coset Spaces. - V Homotopy Theory of CW-complexes. - 1. The Effect on the Homotopy Groups of a Cellular Extension. - 2. Spaces with Prescribed Homotopy Groups. - 3. Weak Homotopy Equivalence and CW-approximation. - 4. Aspherical Spaces. - 5. Obstruction Theory. - 6. Homotopy Extension and Classification Theorems. - 7. Eilenberg-Mac Lane Spaces. - 8. Cohomology Operations. - VI Homology with Local Coefficients. - 1. Bundles of Groups. - 2. Homology with Local Coefficients. - 3. Computations and Examples. - 4. Local Coefficients in CW-complexes. - 5. Obstruction Theory in Fibre Spaces. - 6. The PrimaryObstruction to a Lifting. - 7. Characteristic Classes of Vector Bundles. - VII Homology of Fibre Spaces: Elementary Theory. - 1. Fibrations over a Suspension. - 2. The James Reduced Products. - 3. Further Properties of the Wang Sequence. - 4. Homology of the Classical Groups. - 5. Fibrations Having a Sphere as Fibre. - 6. The Homology Sequence of a Fibration. - 7. The Blakers-Massey Homotopy Excision Theorem. - VIII The Homology Suspension. - 1. The Homology Suspension. - 2. Proof of the Suspension Theorem. - 3. Applications. - 4. Cohomology Operations. - 5. Stable Operations. - 6. The mod 2 Steenrod Algebra. - 7. The Cartan Product Formula. - 8. Some Relations among the Steenrod Squares. - The Action of the Steenrod Algebra on the Cohomology of Some Compact Lie Groups. - IX Postnikov Systems. - 1. Connective Fibrations. - 2. The Postnikov Invariants of a Space. - 3. Amplifying a Space by a Cohomology Class. - 4. Reconstruction of a Space from its Postnikov System. - 5. Some Examples. - 6. Relative Postnikov Systems. - 7. Postnikov Systems and Obstruction Theory. - X On Mappings into Group-like Spaces. - 1. The Category of a Space. - 2. H0-spaces. - 3. Nilpotency of [X G]. - 4. The Case X = X1 × × Xk. - 5. The Samelson Product. - 6. Commutators and Homology. - 7. The Whitehead Product. - 8. Operations in Homotopy Groups. - XI Homotopy Operations. - 1. Homotopy Operations. - 2. The Hopf Invariant. - 3. The Functional Cup Product. - 4. The Hopf Construction. - 5. Geometrical Interpretation of the Hopf Invariant. - 6. The Hilton-Milnor Theorem. - 7. Proof of the Hilton-Milnor Theorem. - 8. The Hopf-Hilton Invariants. - XII Stable Homotopy and Homology. - 1. Homotopy Properties of the James Imbedding. - 2. Suspension and Whitehead Products. - 3. The Suspension Category. - 4. Group Extensions and Homology. - 5. Stable Homotopy as a Homology Theory. - 6. Comparison with the Eilenberg-Steenrod Axioms. - 7. Cohomology Theories. - XIII Homology of Fibre Spaces. - 1. The Homology of a Filtered Space. - 2. Exact Couples. - 3. The Exact Couples of a Filtered Space. - 4. The Spectral Sequence of a Fibration. - 5. Proofs of Theorems (4. 7) and 4. 8). - 6. The Atiyah-Hirzebruch Spectral Sequence. - 7. The Leray-Serre Spectral Sequence. - 8. Multiplicative Properties of the Leray-Serre Spectral Sequence. - 9. Further Applications of the Leray-Serre Spectral Sequence. - Appendix A. - Compact Lie Groups. - 1. Subgroups Coset Spaces Maximal Tori. - 2. Classifying Spaces. - 3. The Spinor Groups. - 6. The Exceptional Jordan Algebra I. - Appendix B. - Additive Relations. - 1. Direct Sums and Products. - 2. Additive Relations. Language: English