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Look Inside Algebraic L-theory and Topological Manifolds

Algebraic L-theory and Topological Manifolds

Part of Cambridge Tracts in Mathematics

  • Date Published: January 2008
  • availability: Available
  • format: Paperback
  • isbn: 9780521055215

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  • 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.

    Reviews & endorsements

    "...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

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    Product details

    • Date Published: January 2008
    • format: Paperback
    • isbn: 9780521055215
    • length: 372 pages
    • dimensions: 229 x 152 x 21 mm
    • weight: 0.55kg
    • availability: Available
  • 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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    Algebraic L-theory and Topological Manifolds

    A. A. Ranicki

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    A. A. Ranicki, University of Edinburgh

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