Classifying Spaces for Surgery and Corbordism of Manifolds. (AM-92), Volume 92 (Annals of Mathematics Studies)

Beginning with a general discussion of bordism, Professors Madsen and Milgram present the homotopy theory of the surgery classifying spaces and the classifying spaces for the various required bundle theories. The next part covers more recent work on the maps between these spaces and the properties of the PL and Top characteristic classes, and includes integrality theorems for topological and PL manifolds. Later chapters treat the integral cohomology of BPL and Btop. The authors conclude with a discussion of the PL and topological cobordism rings and a construction of the torsion-free generators. CONTENTS INTRODUCTION CHAPTER 1. CLASSIFYING SPACES AND COBORDISM A. Bundles with fiber F and structure group II B. The classifying spaces for the classical Lie groups C. The cobordism classification of closed manifolds D. Oriented cobordism theories and localization E. Connections between cobordism and characteristic classes CHAPTER 2. THE SURGERY CLASSIFICATION OF MANIFOLDS A. Poincare duality spaces and the Spivak normal bundle B. The Browder-Novikov theorems and degree 1 normal maps C. The number of manifolds in a homotopy type CHAPTER 3. THE SPACES SG AND BSG A. The spaces of stable homotopy equivalences B. The space Q(S^0) and its structure C. Wreath products, transfer, and the Sylow 2-subgroups of Σn D. A detecting family for the Sylow 2-subgroups of Σn E. The image of H*(BΣn) in the cohomology of the detecting groups F. The homology of Q(S^0) and SG G. The proof of Theorem 3.32 CHAPTER 4. THE HOMOTOPY STRUCTURE OF G/PL AND G/TOP A. The 2-local homotopy type of G/PL B. Ring spectra, orientations and K-theory at odd primes C. Piece-wise linear Pontrjagin classes D. The homotopy type of G/PL[1/2] E. The H-space structure of G/PL CHAPTER 5. THE HOMOTOPY STRUCTURE OF MSPL[1/2] AND MSTOP[1/2] A. The KO-orientation of PL-bundles away from 2 B. The splitting of p-local PL-bundles, p odd C. The homotopy types of G/O[p] and SG[p] D. The splitting of MSPL[p], p odd E. Brumfiel's results F. The map f : SG[p] → BU⊗[p] CHAPTER 6. INFINITE LOOP SPACES AND THEIR HOMOLOGY OPERATIONS A. Homology operations B. Homology operations in H*(Q(S^0)) and H*(SG) C. The Pontrjagin ring H*(BSG) CHAPTER 7. THE 2-LOCAL STRUCTURE OF B(G/TOP) A. Products of Eilenberg-MacLane spaces and operations in H*(G/TOP) B. Massey products in infinite loop spaces C. The proof of Theorem 7.1 CHAPTER 8. THE TORSION FREE STRUCTURE OF THE ORIENTED COBORDISM RINGS A. The map η: Ω*(G/PL) B. The Kervaire and Milnor manifolds C. Constructing the exotic complex projective spaces CHAPTER 9. THE TORSION FREE COHOMOLOGY OF G/TOP AND G/PL A. An important Hopf algebra B. The Hopf algebras F*(BSO⊗) and F*(G/PL) ⊗ Z[1/2] C. The 2-local and integral structure of F*(G/PL) and F*(G/TOP) CHAPTER 10. THE TORSION FREE COHOMOLOGY OF BTOP AND BPL A. The map j* : F*(BO) ⊗ F*(G/TOP) → F*(BTOP) B. The embedding of F*(BTOP; Z(2)) in H*(BTOP; Q) C. The structure of /Tor⊗Z(2) CHAPTER 11. INTEGRALITY THEOREMS A. The inclusion F*(BTOP; Z[1/2]) ⊂ H*(BTOP; Q) B. Piece-wise linear Hattori-Stong theorems C. Milnor's criteria for PL manifolds CHAPTER 12. THE SMOOTH SURGERY CLASSES AND H*(BTOP; Z/2) A. The map B(r×s) : B(G/O) → B^2O×B(G/TOP) B. The Leray-Serre spectral sequence for BTOP CHAPTER 13. THE BOCKSTEIN SPECTRAL SEQUENCE FOR BTOP A. The Bockstein spectral sequences for BO, G/TOP and B(G/O) B. The spectral sequence for BTOP C. The differentials in the subsequence 13.21 CHAPTER 14. THE TYPES OF TORSION GENERATORS A. Torsion generators, suspension and the map η B. Torsion coming from relations involving the Milnor manifolds C. Applications to the structure of the unoriented bordism rings and D. p-torsion in for p odd APPENDIX. THE PROOFS OF 13.12, 13.13, AND 13.15 BIBLIOGRAPHY INDEX