Algebra and Applications

The Local Structure of Algebraic K-Theory

Authors: Dundas, Bjørn Ian, Goodwillie, Thomas G., McCarthy, Randy

  • 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
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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)

Table of contents (7 chapters)

  • Algebraic K-Theory

    Dundas, Bjørn Ian (et al.)

    Pages 1-61

  • Gamma-Spaces and S-Algebras

    Dundas, Bjørn Ian (et al.)

    Pages 63-101

  • Reductions

    Dundas, Bjørn Ian (et al.)

    Pages 103-142

  • Topological Hochschild Homology

    Dundas, Bjørn Ian (et al.)

    Pages 143-178

  • The Trace K→THH

    Dundas, Bjørn Ian (et al.)

    Pages 179-226

Buy this book

eBook $99.00
price for USA (gross)
  • ISBN 978-1-4471-4393-2
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $129.00
price for USA
  • ISBN 978-1-4471-4392-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $129.00
price for USA
  • ISBN 978-1-4471-5904-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
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Bibliographic Information

Bibliographic Information
Book Title
The Local Structure of Algebraic K-Theory
Authors
Series Title
Algebra and Applications
Series Volume
18
Copyright
2012
Publisher
Springer-Verlag London
Copyright Holder
Springer-Verlag London
eBook ISBN
978-1-4471-4393-2
DOI
10.1007/978-1-4471-4393-2
Hardcover ISBN
978-1-4471-4392-5
Softcover ISBN
978-1-4471-5904-9
Series ISSN
1572-5553
Edition Number
1
Number of Pages
XVI, 436
Topics