Author | Whitehead, George W. author |
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Title | Elements of Homotopy Theory [electronic resource] / by George W. Whitehead |
Imprint | New York, NY : Springer New York, 1978 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-6318-0 |
Descript | XXI, 746 p. online resource |
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. Hโ-spaces -- 6. Exact Sequences of Mapping Functors -- 7. Homology Properties of H-spaces and Hโ-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 Primary Obstruction 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 ร ยท ยท ยท ร 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