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Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem


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Part of New Mathematical Monographs

  • Date Published: July 2021
  • availability: Available
  • format: Hardback
  • isbn: 9781108831444

£ 135.00

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About the Authors
  • The long-standing Kervaire invariant problem in homotopy theory arose from geometric and differential topology in the 1960s and was quickly recognised as one of the most important problems in the field. In 2009 the authors of this book announced a solution to the problem, which was published to wide acclaim in a landmark Annals of Mathematics paper. The proof is long and involved, using many sophisticated tools of modern (equivariant) stable homotopy theory that are unfamiliar to non-experts. This book presents the proof together with a full development of all the background material to make it accessible to a graduate student with an elementary algebraic topology knowledge. There are explicit examples of constructions used in solving the problem. Also featuring a motivating history of the problem and numerous conceptual and expository improvements on the proof, this is the definitive account of the resolution of the Kervaire invariant problem.

    • A complete account of the authors' milestone solution to the Kervaire invariant problem in homotopy theory
    • Includes background results in stable homotopy theory that have not previously appeared in the literature
    • Accessible to graduate students with an elementary knowledge of algebraic topology
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    • Winner, 2022 PROSE Award for Mathematics

    Reviews & endorsements

    '… the book succeeds in simultaneously being readable as well as presenting a complex result, in providing tools without being lost in details, and in showing an exciting journey from classical to (at the time of this review) modern stable homotopy theory. Thus, we can expect that it will find a home on many topologists' bookshelves.' Constanze Roitzheim, zbMATH

    'The purpose of the book under review is to give an expanded and systematic development of the part of equivariant stable homotopy theory required by readers wishing to understand the proof of the Kervaire Invariant Theorem. The book fully achieves this design aim. The book ends with a 130-page summary of the proof of the theorem, and having this as a target shapes the entire narrative.' J. P. C. Greenlees, MathSciNet

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    Product details

    • Date Published: July 2021
    • format: Hardback
    • isbn: 9781108831444
    • length: 888 pages
    • dimensions: 250 x 175 x 47 mm
    • weight: 0.164kg
    • availability: Available
  • Table of Contents

    1. Introduction
    Part I. The Categorical Tool Box:
    2. Some Categorical Tools
    3. Enriched Category Theory
    4. Quillen's Theory of Model Categories
    5. Model Category Theory Since Quillen
    6. Bousfield Localization
    Part II. Setting Up Equivariant Stable Homotopy Theory:
    7. Spectra and Stable Homotopy Theory
    8. Equivariant Homotopy Theory
    9. Orthogonal G-spectra
    10. Multiplicative Properties of G-spectra
    Part III. Proving the Kervaire Invariant Theorem:
    11. The Slice Filtration and Slice Spectral Sequence
    12. The Construction and Properties of $MU_{\R}$
    13. The Proofs of the Gap, Periodicity and Detection Theorems
    Table of Notation

  • Authors

    Michael A. Hill, University of California, Los Angeles
    Michael A. Hill is Professor at the University of California, Los Angeles. He is the author of several papers on algebraic topology and is an editor for journals including Mathematische Zeitschrift and Transactions of the American Mathematical Society.

    Michael J. Hopkins, Harvard University, Massachusetts
    Michael J. Hopkins is Professor at Harvard University. His research concentrates on algebraic topology. In 2001, he was awarded the Oswald Veblen Prize in Geometry from the AMS for his work in homotopy theory, followed by the NAS Award in Mathematics in 2012 and the Nemmers Prize in Mathematics in 2014.

    Douglas C. Ravenel, University of Rochester, New York
    Douglas C. Ravenel is the Fayerweather Professor of Mathematics at the University of Rochester. He is the author of two influential previous books in homotopy theory and roughly 75 journal articles on stable homotopy theory and algebraic topology.


    • Winner, 2022 PROSE Award for Mathematics

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