Contents xv Chapter 32. Application: Bott Periodicity 681 §32.1. The Cohomology Algebra of BU(n) 682 §32.2. The Torus and the Symmetric Group 682 §32.3. The Homology Algebra of BU 685 §32.4. The Homology Algebra of ΩSU(n) 689 §32.5. Generating Complexes for ΩSU and BU 690 §32.6. The Bott Periodicity Theorem 692 §32.7. K-Theory 695 §32.8. Additional Problems and Projects 698 Chapter 33. Using the Leray-Serre Spectral Sequence 699 §33.1. The Zeeman Comparison Theorem 699 §33.2. A Rational Borel-Type Theorem 702 §33.3. Mod 2 Cohomology of K(G, n) 703 §33.4. Mod p Cohomology of K(G, n) 706 §33.5. Steenrod Operations Generate Ap 710 §33.6. Homotopy Groups of Spheres 711 §33.7. Spaces Not Satisfying the Ganea Condition 713 §33.8. Spectral Sequences and Serre Classes 714 §33.9. Additional Problems and Projects 716 Part 7. Vistas Chapter 34. Localization and Completion 721 §34.1. Localization and Idempotent Functors 722 §34.2. Proof of Theorem 34.5 726 §34.3. Homotopy Theory of P-Local Spaces 729 §34.4. Localization with Respect to Homology 734 §34.5. Rational Homotopy Theory 737 §34.6. Further Topics 742 Chapter 35. Exponents for Homotopy Groups 745 §35.1. Construction of α 747 §35.2. Spectral Sequence Computations 751 §35.3. The Map γ 754 §35.4. Proof of Theorem 35.3 754 §35.5. Nearly Trivial Maps 756
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