Topology occupies a central position in modern mathematics, and the concept of the fibre bundle provides an appropriate framework for studying differential geometry. Fibrewise homotopy theory is a very large subject that has attracted a good deal of research in recent years. This book provides an overview of the subject as it stands at present.
Table of ContentsI. A Survey of Fibrewise Homotopy Theory.- 1: An Introduction to Fibrewise Homotopy Theory.- 1. Fibrewise spaces.- 2. Fibrewise transformation groups.- 3. Fibrewise homotopy.- 4. Fibrewise cofibrations.- 5. Fibrewise fibrations.- 6. Numerable coverings.- 7. Fibrewise fibre bundles.- 8. Fibrewise mapping-spaces.- 2: The Pointed Theory.- 9. Fibrewise pointed spaces.- 10. Fibrewise one-point (Alexandroff) compactification.- 11. Fibrewise pointed homotopy.- 12. Fibrewise pointed cofibrations.- 13. Fibrewise pointed fibrations.- 14. Numerable coverings (continued).- 15. Fibrewise pointed mapping-spaces.- 16. Fibrewise well-pointed and fibrewise non-degenerate spaces.- 17. Fibrewise complexes.- 18. Fibrewise Whitehead products.- 3: Applications.- 19. Numerical invariants.- 20. The reduced product (James) construction.- 21. Fibrewise Hopf and coHopf structures.- 22. Fibrewise manifolds.- 23. Fibrewise configuration spaces.- II. An Introduction to Fibrewise Stable Homotopy Theory.- 1: Foundations.- 1. Fibre bundles.- 2. Complements on homotopy theory.- 3. Stable homotopy theory.- 4. The Euler class.- 2: Fixed-point Methods.- 5. Fibrewise Euclidean and Absolute Neighbourhood Retracts.- 6. Lefschetz fixed-point theory for fibrewise ENRs.- 7. Fixed-point theory for fibrewise ANRs.- 8. Virtual vector bundles and stable spaces.- 9. The Adams conjecture.- 10. Duality.- 3: Manifold Theory.- 11. Fibrewise differential topology.- 12. The Pontrjagin-Thom construction.- 13. Miller’s stable splitting of U(n).- 14. Configuration spaces and splittings.- 4: Homology Theory.- 15. Fibrewise homology.- References.