Algebraic L-theory and Topological Manifolds

This book explains the applications of quadratic forms to the classification of topological manifolds, in a unified algebraic framework.

A. A. Ranicki (Author)

9780521055215, Cambridge University Press

Paperback / softback, published 21 January 2008

372 pages
22.8 x 15.1 x 2 cm, 0.6 kg

"...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

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 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.

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 ?-? 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.

Subject Areas: Algebraic topology [PBPD]