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OverviewAlgebraic 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. Full Product DetailsAuthor: Bjørn Ian Dundas , Thomas G. Goodwillie , Randy McCarthyPublisher: Springer London Ltd Imprint: Springer London Ltd Edition: 2012 ed. Volume: 18 Dimensions: Width: 15.50cm , Height: 2.50cm , Length: 23.50cm Weight: 0.840kg ISBN: 9781447143925ISBN 10: 1447143922 Pages: 436 Publication Date: 06 September 2012 Audience: Professional and scholarly , Professional & Vocational Format: Hardback Publisher's Status: Active Availability: Manufactured on demand We will order this item for you from a manufactured on demand supplier. Table of ContentsAlgebraic K-theory.- Gamma-spaces and S-algebras.- Reductions.- Topological Hochschild Homology.- The Trace K → THH.- Topological Cyclic Homology.- The Comparison of K-theory and TC.ReviewsFrom 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) 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) Author InformationTab Content 6Author Website:Countries AvailableAll regions |