Algebraic L-theory and Topological Manifolds

Algebraic L-theory and Topological Manifolds

by A. A. Ranicki
     
 

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ISBN-10: 0521055210

ISBN-13: 9780521055215

Pub. Date: 03/28/2008

Publisher: Cambridge University Press

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

Overview

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.

Product Details

ISBN-13:
9780521055215
Publisher:
Cambridge University Press
Publication date:
03/28/2008
Series:
Cambridge Tracts in Mathematics Series, #102
Pages:
372
Product dimensions:
5.98(w) x 8.98(h) x 0.83(d)

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.

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