Algebraic L-theory and Topological Manifolds
Series: Cambridge Tracts in Mathematics; 102;
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Product details:
- Publisher Cambridge University Press
- Date of Publication 10 December 1992
- ISBN 9780521420242
- Binding Hardback
- No. of pages372 pages
- Size 236x159x23 mm
- Weight 655 g
- Language English 0
Categories
Short description:
This book explains the applications of quadratic forms to the classification of topological manifolds, in a unified algebraic framework.
MoreLong description:
This book presents a definitive account of the applications of the algebraic L-theory to the surgery classification of topological manifolds. The central result is the identification of a manifold structure in the homotopy type of a Poincar&&&233; 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 Poincar&&&233; 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. The book is designed as an introduction to the subject, accessible to graduate students in topology; no previous acquaintance with surgery theory is assumed, and every algebraic concept is justified by its occurrence in topology.
"...develops lower K- and L-theory with a view to applications in topology....Apart from the obvious interest of this text both to topologists and to K-theorists, it also serves as an introduction to the field, since there is a comprehensive survey of previous results and applications." M.E. Keating, Bulletin of the London Mathematical Society
Table of Contents:
Introduction; Summary; Part I. Algebra: 1. Algebraic Poincar&&&233; complexes; 2. Algebraic normal complexes; 3. Algebraic bordism categories; 4. Categories over complexes; 5. Duality; 6. Simply connected assembly; 7. Derived product and Hom; 8. Local Poincar&&&233; duality; 9. Universal assembly; 10. The algebraic &&&960;-&&&960; theorem; 11. &&&8710;-sets; 12. Generalized 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 Poincar&&&233; complexes; 20. The simply connected case; 21. Transfer; 22. Finite fundamental group; 23. Splitting; 24. Higher signatures; 25. The 4-periodic theory; 26. Surgery with coefficients; Appendices; Bibliography; Index.
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