Elements of Homotopy Theory

As the title suggests, this book is concerned with the elementary portion of the subject of homotopy theory. It is assumed that the reader is familiar with the fundamental group and with singular homology theory, including the Universal Coefficient and Kiinneth Theorems. Some acquaintance with manifolds and Poincare duality is desirable, but not essential. Anyone who has taught a course in algebraic topology is familiar with the fact that a formidable amount of technical machinery must be introduced and mastered before the simplest applications can be made. This phenomenon is also observable in the more advanced parts of the subject. I have attempted to short-circuit it by making maximal use of elementary methods. This approach entails a leisurely exposition in which brevity and perhaps elegance are sacrificed in favor of concreteness and ease of application. It is my hope that this approach will make homotopy theory accessible to workers in a wide range of other subjects-subjects in which its impact is beginning to be felt. It is a consequence of this approach that the order of development is to a certain extent historical. Indeed, if the order in which the results presented here does not strictly correspond to that in which they were discovered, it nevertheless does correspond to an order in which they might have been discovered had those of us who were working in the area been a little more perspicacious.

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 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 × · · · × 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.