Synopses & Reviews
This book presents the definitive account of the applications of this algebra to the surgery classification of topological manifolds. The central result is the identification of a manifold structure in the homotopy type of a Poincaré duality space with a local quadratic structure in the chain homotopy type of the universal cover. The difference between the homotopy types of manifolds and Poincaré duality spaces is identified with the fibre of the algebraic L-theory assembly map, which passes from local to global quadratic duality structures on chain complexes. The algebraic L-theory assembly map is used to give a purely algebraic formulation of the Novikov conjectures on the homotopy invariance of the higher signatures; any other formulation necessarily factors through this one.
This book explains the applications of quadratic forms to the classificati
This book explains the applications of quadratic forms to the classification of topological manifolds, in a unified algebraic framework.
Surgery theory is the main technique for classifying high-dimensional topological manifolds. Algebraic L-theory is the name of the abstract algebra used to study their geometric properties; it involves the study of quadratic forms. Assuming no previous acquaintance with surgery theory and justifying all the algebraic concepts used by their relevance to topology, Dr Ranicki explains the applications of quadratic forms to the classification of topological manifolds, in a unified algebraic framework.
on of topological manifolds, in a unified algebraic framework.
Table of Contents
Introduction; Part I. Algebra: 1. Algebraic Poincarécomplexes; 2. Algebraic normal complexes; 3. Algebraic normal categories; 4. Categories over complexes; 5. Duality; 6. Simply-connected assembly; 7. Derived products and Hom; 8. Local Poincaréduality; 9. Universal assembly; 10. The algebraic -theorem; 11. -sets; 12. Generalised homology theory; 13. Algebraic L-spectra; 14. The algebraic surgery exact sequence; 15. Connective L-theory; Part II. Topology: 16. The L-theory orientation of topology; 17. The total surgery obstruction; 18. The structure set; 19. Geometric Poincarécomplexes; 20. The simply-connected case; 21. Transfer; 22. Finite fundamental group; 23. Splitting; 24. Higher signatures; 25. Periodicity; 26. Surgery with coefficients; Appendices; References; Index.