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Fibrewise Homotopy Theory

  • Michael Charles Crabb
  • Ioan Mackenzie James

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

Table of contents

  1. Front Matter
    Pages i-viii
  2. A Survey of Fibrewise Homotopy Theory

    1. Front Matter
      Pages 1-2
    2. Michael Charles Crabb, Ioan Mackenzie James
      Pages 3-51
    3. Michael Charles Crabb, Ioan Mackenzie James
      Pages 53-99
    4. Michael Charles Crabb, Ioan Mackenzie James
      Pages 101-135
  3. An Introduction to Fibrewise Stable Homotopy Theory

    1. Front Matter
      Pages 137-140
    2. Michael Charles Crabb, Ioan Mackenzie James
      Pages 141-185
    3. Michael Charles Crabb, Ioan Mackenzie James
      Pages 187-242
    4. Michael Charles Crabb, Ioan Mackenzie James
      Pages 243-308
    5. Michael Charles Crabb, Ioan Mackenzie James
      Pages 309-329
  4. Back Matter
    Pages 331-341

About this book

Introduction

Topology occupies a central position in the mathematics of today. One of the most useful ideas to be introduced in the past sixty years is the concept of fibre bundle, which provides an appropriate framework for studying differential geometry and much else. Fibre bundles are examples of the kind of structures studied in fibrewise topology. Just as homotopy theory arises from topology, so fibrewise homotopy the­ ory arises from fibrewise topology. In this monograph we provide an overview of fibrewise homotopy theory as it stands at present. It is hoped that this may stimulate further research. The literature on the subject is already quite extensive but clearly there is a great deal more to be done. Efforts have been made to develop general theories of which ordinary homotopy theory, equivariant homotopy theory, fibrewise homotopy theory and so forth will be special cases. For example, Baues [7] and, more recently, Dwyer and Spalinski [53], have presented such general theories, derived from an earlier theory of Quillen, but none of these seem to provide quite the right framework for our purposes. We have preferred, in this monograph, to develop fibre wise homotopy theory more or less ab initio, assuming only a basic knowledge of ordinary homotopy theory, at least in the early sections, but our aim has been to keep the exposition reasonably self-contained.

Keywords

Homotopy differential geometry homology homotopy theory manifold topology

Authors and affiliations

  • Michael Charles Crabb
    • 1
  • Ioan Mackenzie James
    • 2
  1. 1.Department of MathematicsUniversity of AberdeenAberdeenUK
  2. 2.Mathematical InstituteOxford UniversityOxfordUK

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4471-1265-5
  • Copyright Information Springer-Verlag London 1998
  • Publisher Name Springer, London
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4471-1267-9
  • Online ISBN 978-1-4471-1265-5
  • Series Print ISSN 1439-7382
  • Buy this book on publisher's site