Algebraic L-theory and Topological Manifolds
Algebraic L-theory and Topological Manifolds
£54.00
GBPThis 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 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. 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.
Product details
March 2008Paperback
9780521055215
372 pages
228 × 151 × 20 mm
0.6kg
Available
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Table of Contents
- Introduction
- Summary
- Part I. Algebra:
- 1. Algebraic Poincaré 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 Poincaré duality
- 9. Universal assembly
- 10. The algebraic π-π theorem
- 11. ∆-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 Poincaré 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.
