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Ends of Complexes

Part of Cambridge Tracts in Mathematics

  • Date Published: January 2008
  • availability: Available
  • format: Paperback
  • isbn: 9780521055192

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  • The ends of a topological space are the directions in which it becomes noncompact by tending to infinity. The tame ends of manifolds are particularly interesting, both for their own sake, and for their use in the classification of high-dimensional compact manifolds. The book is devoted to the related theory and practice of ends, dealing with manifolds and CW complexes in topology and chain complexes in algebra. The first part develops a homotopy model of the behavior at infinity of a noncompact space. The second part studies tame ends in topology. The authors show tame ends to have a uniform structure, with a periodic shift map. They use approximate fibrations to prove that tame manifold ends are the infinite cyclic covers of compact manifolds. The third part translates these topological considerations into an appropriate algebraic context, relating tameness to homological properties and algebraic K- and L-theory. This book will appeal to researchers in topology and geometry.

    • Ranicki is a well-known author
    • This book ties up all the loose ends of the subject
    • Features a blend of geometric topology, algebraic topology and algebra, designed to appeal to readers with background in only one of these disciplines
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    Reviews & endorsements

    'The book gathers together the main strands of the theory of ends of manifolds from the last thirty years and presents a unified and coherent treatment of them. It also contains authoritative expositions of certain topics in topology such as mapping tori and telescopes, often omitted from textbooks. It is thus simultaneously a research monograph and a useful reference.' Proceedings of the Edinburgh Mathematical Society

    'This is a highly specialized monograph which is very clearly written and made as accessible for the reader as possible … It is absolutely indispensable for any specialist in the field.' European Mathematical Society

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    Product details

    • Date Published: January 2008
    • format: Paperback
    • isbn: 9780521055192
    • length: 380 pages
    • dimensions: 228 x 151 x 21 mm
    • weight: 0.576kg
    • availability: Available
  • Table of Contents

    Introduction
    Chapter summaries
    Part I. Topology at Infinity:
    1. End spaces
    2. Limits
    3. Homology at infinity
    4. Cellular homology
    5. Homology of covers
    6. Projective class and torsion
    7. Forward tameness
    8. Reverse tameness
    9. Homotopy at infinity
    10. Projective class at infinity
    11. Infinite torsion
    12. Forward tameness is a homotopy pushout
    Part II. Topology Over the Real Line:
    13. Infinite cyclic covers
    14. The mapping torus
    15. Geometric ribbons and bands
    16. Approximate fibrations
    17. Geometric wrapping up
    18. Geometric relaxation
    19. Homotopy theoretic twist glueing
    20. Homotopy theoretic wrapping up and relaxation
    Part III. The Algebraic Theory:
    21. Polynomial extensions
    22. Algebraic bands
    23. Algebraic tameness
    24. Relaxation techniques
    25. Algebraic ribbons
    26. Algebraic twist glueing
    27. Wrapping up in algebraic K- and L-theory
    Part IV. Appendices
    References
    Index.

  • Resources for

    Ends of Complexes

    Bruce Hughes, Andrew Ranicki

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  • Authors

    Bruce Hughes, Vanderbilt University, Tennessee

    Andrew Ranicki, University of Edinburgh

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