The cyclotomic trace and algebraic K-theory of spaces

@article{Bkstedt1993TheCT,
  title={The cyclotomic trace and algebraic K-theory of spaces},
  author={Marcel B{\"o}kstedt and Wu-chung Hsiang and Ib Henning Madsen},
  journal={Inventiones mathematicae},
  year={1993},
  volume={111},
  pages={465-539}
}
The cyclotomic trace is a map from algebraic K-theory of a group ring to a certain topological refinement of cyclic homology. The target is naturally mapped to topological Hochschild homology, and the cyclotomic trace lifts the topological Dennis trace. Our cyclic homology can be defined also for "group rings up to homotopy", and in this setting the cyclotomic trace produces invariants of Waldhausen's A-theory. Our main applications go in two directions. We show on the one hand that the K… 

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References

SHOWING 1-10 OF 38 REFERENCES

Cyclic homology and equivariant homology

The purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of Connes [9-11], see also Loday and Quillen [20], and "IF equivariant homology and

On axiomatic homology theory.

provide a protective representation of H(X) as a direct product. It is easily verified that the singular homology and cohomology theories are additive. Also the Cech theories based on infinite

Equivariant Stable Homotopy Theory

The last decade has seen a great deal of activity in this area. The chapter provides a brief sketch of the basic concepts of space-level equivariant homotopy theory. It also provides an introduction

equivariant function spaces and characteristic classes

We determine the structure of the homology of the BeckerSchultz space SGiS1) ~ Q(CP^° AS1) of stable Sx-equivariant self-maps of spheres (with standard free S1 -action) as a Hopf algebra over the

On the general linear group and Hochschild homology

Our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring. Before stating the result we establish some notation

Configuration-spaces and iterated loop-spaces

The object of this paper is to prove a theorem relating "configurationspaces" to iterated loop-spaces. The idea of the connection between them seems to be due to Boardman and Vogt [2]. Part of the

Infinite Loop G-Spaces Associated to Monoidal G-Graded Categories

We construct a functor KG which takes each pair of monoidal G-graded categories (D,Df) to an infinite loop G-space KG(D,D'). When D'=D, its homotopy groups n%KG(D,D) coincide with the equivariant

Prerequisites (on equivariant stable homotopy) for Carlssons's lecture

Cambridge CB2 ISB ENGLAND §i. Introduction. Three things might be done to help those who wish to understand Carlsson's work on Seqal's Burnside Ring Conjecture [8]. First, one might attempt a general

Homotopy Limits, Completions and Localizations

Completions and localizations.- The R-completion of a space.- Fibre lemmas.- Tower lemmas.- An R-completion of groups and its relation to the R-completion of spaces.- R-localizations of nilpotent

The geometry of iterated loop spaces

Operads and -spaces.- Operads and monads.- A? and E? operads.- The little cubes operads .- Iterated loop spaces and the .- The approximation theorem.- Cofibrations and quasi-fibrations.- The smash