Trace maps from the algebraic \(K\)-theory of the integers (after Marcel Bökstedt) (Q1380056)

Let \(K({\mathbb{Z}})\) be the \(K\)-theory spectrum and \(T({\mathbb{Z}} )\) the topological Hochschild homology spectrum of the integers and set \(K_i({\mathbb{Z}} ):=\pi_iK({\mathbb{Z}})\) and \(T_i({\mathbb{Z}}):=\pi_iT({\mathbb{Z}})\). By Bökstedt's calculations: \(T_0({\mathbb{Z}})={\mathbb{Z}}\) and \(T_{2i-1}({\mathbb{Z}})\cong {\mathbb{Z}}/i\) for all \(i\in{\mathbb{N}}\), while the remaining groups are zero [\textit{M. Bökstedt}, ``The topological Hochschild homology of \({\mathbb{Z}}\) and \({\mathbb{Z}}/p\)'', Ann. of Math., II. Ser., to appear]. The goal of this article is to provide a reference for two other results by \textit{M. Bökstedt} which had remained unpublished. The first reads: let \(p\) be any prime. Bökstedt's trace map induces a surjection \(K_{2p-1}({\mathbb{Z}})\to T_{2p-1}({\mathbb{Z}})\cong {\mathbb{Z}}/p\). When \(p\) is odd, a proof of this result was published in [\textit{M. Bökstedt} and \textit{I. Madsen}, ``Topological cyclic homology of the integers'', Asterisque 226, 57-143 (1994; Zbl 0816.19001)]. The second result is concerned with detecting elements in the stable homotopy groups of spheres. Bökstedt's map admits a refinement \(K({\mathbb{Z}})_p\to T({\mathbb{Z}})_p^{hS^1},\) (where the subscript \(p\) stands for \(p\)-adic completion and \(T({\mathbb{Z}})^{hS^1}\) for the \(S^1\)-homotopy fixed points of \(T({\mathbb{Z}})\)). Hence a map from \(Q(S^0)_p\) to \(T({\mathbb{Z}})_p^{hS^1}.\) Using this map, it is shown that the first \(p\)-torsion element in degree \(2p-3\) of the stable homotopy groups of spheres is detected in the homotopy of \(T({\mathbb{Z}})_p^{hS^1}\).