The cyclotomic trace and algebraic K-theory of spaces
type: Item
instance of: scholarly article
title: The cyclotomic trace and algebraic K-theory of spaces
zbMATH Open document ID: 0804.55004
DOI: 10.1007/BF01231296
published in: Inventiones Mathematicae
publication date: 1993-06-17T00:00:00Z
full work available at URL: https://eudml.org/doc/144086
review text: For a group \(G\) one may embed \(BG\) as constant loops into \({\mathcal C} (S^ 1, BG):= \Lambda BG\). The map actually has its target in \(\lim_ \leftarrow (\Lambda BG)^{C_{p^ n}}\). \(C_{p^ n}\) denotes the cyclic group of order \(p^ n\). The limit is taken over inclusions of fixed point sets. There is a self map \(\Phi_ p\) of the inverse system, given by calculating the \(C_{p^{n+1}}\) fixed-point set from the \(C_{p^ n}\) fixed set. This observation yields in the homotopy invariant setting (\(G\) a grouplike monoid) considered by the authors the map \({\mathcal F}: BG\to (\text{holim}_ \leftarrow (\Lambda BG)^{C_{p^ n}})^{h \Phi_ p}\), where \(h\Phi_ p\) is the homotopy equalizer of \(\Phi_ p\) and the identity. The map becomes derived from a simplicial approach, which describes the \(S^ 1\)- equivariant homotopy type of \(\Lambda BG\) as the realization of the cyclic bar construction \(N_ \bullet^{cy} (G)\) on the fixed-point sets of finite subgroups (!) of \(S^ 1\) (Proposition 2.6). In case \(G= \text{Gl}_ n(R)\), the invertible matrices over a ring \(R\), the cyclic bar construction on \(\text{Gl}_ n(R)\) has a trace in \(N^{cy}_{\otimes;\bullet} (M_ n(R))\), \(M_ n(R)\) all \(n\times n\) matrices, which identifies by Morita invariance with \(N^{cy}_{\otimes;\bullet} (R)\), that is the Hochschild homology of \(R\). The first author's construction [``Topological Hochschild homology'', Topology (to appear)] promotes this relation to rings up to homotopy, resulting in the invention of topological Hochschild homology \(\text{THH}(R)\). The authors push the construction of the map \({\mathcal F}\) through this pipe also, eventually refining the Dennis trace to an infinite loop map (Definition 5.12) \({\mathcal F}: K(R)\to (\text{holim}_ \leftarrow \text{THH} (R)^{C^ n_ p})^{h\Phi_ p}=: \text{TC}(R,p)\). The (new) functor is called the topological cyclic homology of \(R\) (at \(p\)) and \({\mathcal F}\) becomes renamed Trc. The major part of the paper then prove…
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 19D10
Mathematics Subject Classification ID: 55N35
Mathematics Subject Classification ID: 57R52
Mathematics Subject Classification ID: 55P60
zbMATH DE Number: 203552
zbMATH Keywords: free loop space of a classifying space
zbMATH Keywords: fixed-point sets of finite subgroups of \(S^ 1\)
zbMATH Keywords: groups with finitely generated integral homology
zbMATH Keywords: cyclic bar construction
zbMATH Keywords: invertible matrices over a ring
zbMATH Keywords: rings up to homotopy
zbMATH Keywords: topological Hochschild homology
zbMATH Keywords: Dennis trace
zbMATH Keywords: infinite loop map
zbMATH Keywords: topological cyclic homology
zbMATH Keywords: invariants for \(K\)-theory
zbMATH Keywords: Soulé embedding
zbMATH Keywords: Diff-Whitehead space
zbMATH Keywords: assembly map
zbMATH Keywords: Waldhausen \(K\)-theory
author: Marcel Bökstedt
author: Wu-Chung Hsiang
author: Ib Madsen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 203552