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The Geometric Hopf Invariant and Surgery Theory

  • Provides the homotopy theoretic foundations for surgery theory

  • Includes a self-contained account of the Hopf invariant in terms of Z_2-equivariant homotopy

  • Covers applications of the Hopf invariant to surgery theory, in particular the Double Point Theorem


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

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Michael Crabb, Andrew Ranicki
    Pages 1-15
  3. Michael Crabb, Andrew Ranicki
    Pages 17-38
  4. Michael Crabb, Andrew Ranicki
    Pages 39-75
  5. Michael Crabb, Andrew Ranicki
    Pages 127-208
  6. Michael Crabb, Andrew Ranicki
    Pages 209-295
  7. Michael Crabb, Andrew Ranicki
    Pages 297-304
  8. Michael Crabb, Andrew Ranicki
    Pages 305-327
  9. Back Matter
    Pages 329-397

About this book


Written by leading experts in the field, this monograph provides homotopy theoretic foundations for surgery theory on higher-dimensional manifolds.

Presenting classical ideas in a modern framework, the authors carefully highlight how their results relate to (and generalize) existing results in the literature. The central result of the book expresses algebraic surgery theory in terms of the geometric Hopf invariant, a construction in stable homotopy theory which captures the double points of immersions. Many illustrative examples and applications of the abstract results are included in the book, making it of wide interest to topologists.

Serving as a valuable reference, this work is aimed at graduate students and researchers interested in understanding how the algebraic and geometric topology fit together in the surgery theory of manifolds. It is the only book providing such a wide-ranging historical approach to the Hopf invariant, double points and surgery theory, with many results old and new. 


MSC (2010): 55Q25, 57R42 geometric Hopf invariant manifolds doube points of maps double point theorem algebraic surgery difference construction homotopy difference construction chain homotopy coordinate-free approach to stable homotopy theory inner product spaces stable homotopy theory Z_2 equivariant homotopy bordism theory surgery obstruction theory

Authors and affiliations

  1. 1.Institute of MathematicsUniversity of AberdeenAberdeenUnited Kingdom
  2. 2.School of MathematicsUniversity of EdinburghEdinburghUnited Kingdom

Bibliographic information