Elements of Homotopy Theory

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.