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High-dimensional Knot Theory

Algebraic Surgery in Codimension 2

  • Book
  • © 1998

Overview

Authors:
  1. Andrew Ranicki

    1. Department of Mathematics and Statistics, University of Edinburgh, Edinburgh, Scotland, UK

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Part of the book series: Springer Monographs in Mathematics (SMM)

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Table of contents (42 chapters)

  1. Front Matter

    Pages I-XXXVI
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  2. Algebraic K-theory

    1. Front Matter

      Pages 1-1
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    2. Finite structures

      • Andrew Ranicki
      Pages 3-10

    3. Geometric bands

      • Andrew Ranicki
      Pages 11-13

    4. Algebraic bands

      • Andrew Ranicki
      Pages 15-18

    5. Localization and completion in K-theory

      • Andrew Ranicki
      Pages 19-26

    6. K-theory of polynomial extensions

      • Andrew Ranicki
      Pages 27-40

    7. K-theory of formal power series

      • Andrew Ranicki
      Pages 41-46

    8. Algebraic transversality

      • Andrew Ranicki
      Pages 47-56

    9. Finite domination and Novikov homology

      • Andrew Ranicki
      Pages 57-70

    10. Noncommutative localization

      • Andrew Ranicki
      Pages 71-78

    11. Endomorphism K-theory

      • Andrew Ranicki
      Pages 79-96

    12. The characteristic polynomial

      • Andrew Ranicki
      Pages 97-104

    13. Primary K-theory

      • Andrew Ranicki
      Pages 105-118

    14. Automorphism K-theory

      • Andrew Ranicki
      Pages 119-134

    15. Witt vectors

      • Andrew Ranicki
      Pages 135-150

    16. The fibering obstruction

      • Andrew Ranicki
      Pages 151-158

    17. Reidemeister torsion

      • Andrew Ranicki
      Pages 159-164

    18. Alexander polynomials

      • Andrew Ranicki
      Pages 165-176

    19. K-theory of Dedekind rings

      • Andrew Ranicki
      Pages 177-184
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Keywords

About this book

High-dimensional knot theory is the study of the embeddings of n-dimensional manifolds in (n+2)-dimensional manifolds, generalizing the traditional study of knots in the case n=1. The main theme is the application of the author's algebraic theory of surgery to provide a unified treatment of the invariants of codimension 2 embeddings, generalizing the Alexander polynomials and Seifert forms of classical knot theory. Many results in the research literature are thus brought into a single framework, and new results are obtained. The treatment is particularly effective in dealing with open books, which are manifolds with codimension 2 submanifolds such that the complement fibres over a circle. The book concludes with an appendix by E. Winkelnkemper on the history of open books.

Authors and Affiliations

  • Department of Mathematics and Statistics, University of Edinburgh, Edinburgh, Scotland, UK

    Andrew Ranicki

Bibliographic Information

  • Book Title: High-dimensional Knot Theory

  • Book Subtitle: Algebraic Surgery in Codimension 2

  • Authors: Andrew Ranicki

  • Series Title: Springer Monographs in Mathematics

  • DOI: https://doi.org/10.1007/978-3-662-12011-8

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1998

  • Hardcover ISBN: 978-3-540-63389-1Published: 06 August 1998

  • Softcover ISBN: 978-3-642-08329-7Published: 15 December 2010

  • eBook ISBN: 978-3-662-12011-8Published: 17 April 2013

  • Series ISSN: 1439-7382

  • Series E-ISSN: 2196-9922

  • Edition Number: 1

  • Number of Pages: XXXVI, 646

  • Topics: Algebraic Topology

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