Syzygies and Homotopy Theory

  • F.E.A. Johnson

Part of the Algebra and Applications book series (AA, volume 17)

Table of contents

  1. Front Matter
    Pages I-XXIV
  2. Theory

    1. Front Matter
      Pages 1-1
    2. F. E. A. Johnson
      Pages 3-12
    3. F. E. A. Johnson
      Pages 13-36
    4. F. E. A. Johnson
      Pages 37-62
    5. F. E. A. Johnson
      Pages 63-88
    6. F. E. A. Johnson
      Pages 89-116
    7. F. E. A. Johnson
      Pages 117-128
    8. F. E. A. Johnson
      Pages 129-149
    9. F. E. A. Johnson
      Pages 151-163
  3. Practice

    1. Front Matter
      Pages 165-165
    2. F. E. A. Johnson
      Pages 167-173
    3. F. E. A. Johnson
      Pages 175-183
    4. F. E. A. Johnson
      Pages 185-197
    5. F. E. A. Johnson
      Pages 199-212
    6. F. E. A. Johnson
      Pages 213-220
    7. F. E. A. Johnson
      Pages 221-226
    8. F. E. A. Johnson
      Pages 227-238
    9. F. E. A. Johnson
      Pages 239-254
    10. F. E. A. Johnson
      Pages 255-263
  4. Back Matter
    Pages 265-294

About this book


The most important invariant of a topological space is its fundamental group. When this is trivial, the resulting homotopy theory is well researched and familiar. In the general case, however, homotopy theory over nontrivial fundamental groups is much more problematic and far less well understood.

Syzygies and Homotopy Theory explores the problem of nonsimply connected homotopy in the first nontrivial cases and presents, for the first time, a systematic rehabilitation of Hilbert's method of syzygies in the context of non-simply connected homotopy theory. The first part of the book is theoretical, formulated to allow a general finitely presented group as a fundamental group. The innovation here is to regard syzygies as stable modules rather than minimal modules. Inevitably this forces a reconsideration of the problems of noncancellation; these are confronted in the second, practical, part of the book. In particular, the second part of the book considers how the theory works out in detail for the specific examples Fn ´F where Fn is a free group of rank n and F is finite. Another innovation is to parametrize the first syzygy in terms of the more familiar class of stably free modules. Furthermore, detailed description of these stably free modules is effected by a suitable modification of the method of Milnor squares.

The theory developed within this book has potential applications in various branches of algebra, including homological algebra, ring theory and K-theory. Syzygies and Homotopy Theory will be of interest to researchers and also to graduate students with a background in algebra and algebraic topology.


D(2) problem Milnor squares R(2) problem generalized Swan module stable module syzygy

Authors and affiliations

  • F.E.A. Johnson
    • 1
  1. 1.Department of MathematicsUniversity College LondonLondonUnited Kingdom

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag London Limited 2012
  • Publisher Name Springer, London
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4471-2293-7
  • Online ISBN 978-1-4471-2294-4
  • Series Print ISSN 1572-5553
  • Series Online ISSN 2192-2950
  • Buy this book on publisher's site