The cyclotomic trace and algebraic K-theory of spaces. (English) Zbl 0804.55004

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 proves the construction as providing (highly) nontrivial invariants for \(K\)-theory. The authors develop a homotopy- theoretic version of the Soulé embedding [C. Soulé, Invent. Math. 55, No. 3, 251-295 (1979; Zbl 0437.12008); Lect. Notes Math. 854, 372-401 (1981; Zbl 0488.12008)], culminating in Corollary 9.12: \(\text{Trc}: Wh(*)^ \wedge_ p\to Q(\Sigma BO(2))^ \wedge_ p\) is split surjective for odd regular primes (here \(Wh(*)\) is the Diff- Whitehead space in the sense of F. Waldhausen [in ‘Algebraic topology and algebraic \(K\)-theory’, Proc. Conf., Princeton, NJ, Ann. Math. Stud. 113, 392-417 (1987; Zbl 0708.19001)] and \((-)^ \wedge_ p\) is the \(p\)-adic completion). The assembly map \(B\Gamma_ + \wedge A(*)\to A(B\Gamma)\) is shown as rationally split injective (Theorem 9.13) for a class of groups including those with finitely generated integral homology. (The functor \(A\) here of course denotes Waldhausen \(K\)-theory).


55N15 Topological \(K\)-theory
55P47 Infinite loop spaces
19D10 Algebraic \(K\)-theory of spaces
55N35 Other homology theories in algebraic topology
57R52 Isotopy in differential topology
55P60 Localization and completion in homotopy theory
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