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The Local Structure of Algebraic K-Theory

  • Book
  • © 2012

Overview

Authors:
  1. Bjørn Ian Dundas

    1. , Department of Mathematics, University of Bergen, Bergen, Norway

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  2. Thomas G. Goodwillie

    1. , Mathematics Department, Brown University, Providence, USA

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    You can also search for this author in PubMed   Google Scholar

  3. Randy McCarthy

    1. , Department of Mathematics, University of Illinois, Urbana, USA

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  • Covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology, and gives a proof of the fact that the difference between the theories are 'locally constant'
  • Provides an inroad to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework
  • Contains the proof of the integral Goodwillie ICM 1990 conjecture and explains the mathematical prerequisites needed to do this
  • Includes supplementary material: sn.pub/extras
  • Includes supplementary material: sn.pub/extras

Part of the book series: Algebra and Applications (AA, volume 18)

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Table of contents (7 chapters)

  1. Front Matter

    Pages I-XV
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  2. Algebraic K-Theory

    • Bjørn Ian Dundas, Thomas G. Goodwillie, Randy McCarthy
    Pages 1-61

  3. Gamma-Spaces and S-Algebras

    • Bjørn Ian Dundas, Thomas G. Goodwillie, Randy McCarthy
    Pages 63-101

  4. Reductions

    • Bjørn Ian Dundas, Thomas G. Goodwillie, Randy McCarthy
    Pages 103-142

  5. Topological Hochschild Homology

    • Bjørn Ian Dundas, Thomas G. Goodwillie, Randy McCarthy
    Pages 143-178

  6. The Trace K→THH

    • Bjørn Ian Dundas, Thomas G. Goodwillie, Randy McCarthy
    Pages 179-226

  7. Topological Cyclic Homology

    • Bjørn Ian Dundas, Thomas G. Goodwillie, Randy McCarthy
    Pages 227-279

  8. The Comparison of K-Theory and TC

    • Bjørn Ian Dundas, Thomas G. Goodwillie, Randy McCarthy
    Pages 281-332

  9. Back Matter

    Pages 333-435
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About this book

Algebraic K-theory encodes important invariants for several mathematical disciplines, spanning from geometric topology and functional analysis to number theory and algebraic geometry. As is commonly encountered, this powerful mathematical object is very hard to calculate. Apart from Quillen's calculations of finite fields and Suslin's calculation of algebraically closed fields, few complete calculations were available before the discovery of homological invariants offered by motivic cohomology and topological cyclic homology. This book covers the connection between algebraic K-theory and Bökstedt, Hsiang and Madsen's topological cyclic homology and proves that the difference between the theories are ‘locally constant’. The usefulness of this theorem stems from being more accessible for calculations than K-theory, and hence a single calculation of K-theory can be used with homological calculations to obtain a host of ‘nearby’ calculations in K-theory. For instance, Quillen's calculation of the K-theory of finite fields gives rise to Hesselholt and Madsen's calculations for local fields, and Voevodsky's calculations for the integers give insight into the diffeomorphisms of manifolds. In addition to the proof of the full integral version of the local correspondence between K-theory and topological cyclic homology, the book provides an introduction to the necessary background in algebraic K-theory and highly structured homotopy theory; collecting all necessary tools into one common framework. It relies on simplicial techniques, and contains an appendix summarizing the methods widely used in the field. The book is intended for graduate students and scientists interested in algebraic K-theory, and presupposes a basic knowledge of algebraic topology.

Reviews

From the reviews:

“The comparison of K-theory with topological cyclic homology is by means of a natural transformation called the cyclotomic trace which is the principal subject of this book. … Many references invite to further reading. The book can be highly recommended to anybody interested in the modern understanding of algebraic K-theory and its approximations by functors which are more accessible to calculations.” (Rainer Vogt, zbMATH, Vol. 1272, 2013)

Authors and Affiliations

  • , Department of Mathematics, University of Bergen, Bergen, Norway

    Bjørn Ian Dundas

  • , Mathematics Department, Brown University, Providence, USA

    Thomas G. Goodwillie

  • , Department of Mathematics, University of Illinois, Urbana, USA

    Randy McCarthy

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