Homology operations in the topological cyclic homology of a point
type: Item
instance of: scholarly article
title: Homology operations in the topological cyclic homology of a point
zbMATH Open document ID: 1204.55014
author: Håkon Schad Bergsaker
author: John Rognes
published in: Geometry \& Topology
publication date: 2010-02-23T00:00:00Z
review text: Topological Cyclic homology of a point, \(TC(*;p)\) has a commutative \(\mathbb{S}\)-algebra structure. Let \(C_{p^n}\) be the cyclic subgroup of order \(p^n\) of the circle group \(\mathbb{T}\). Then \(TC(*;p)\) is the homotopy inverse limit of a diagram involving the Frobenius and restriction maps \( R,F: \mathbb{S}^{C_{p^n}}\to\mathbb{S}^{C_{p^{n-1}}}\), where \( \mathbb{S}^{C_{p^n}}\) is the \(C_{p^n}\) fixed points of the \(\mathbb{T}\) equivariant sphere spectrum. These spectra are all \(E_{\infty}\) ring spectra. With \(TC^{(1)}(*,p)\) the homotopy inverse limit of \(R,F: \mathbb{S}^{C_{p}}\to \mathbb{S}\), we get canonical maps of \(E_{\infty}\) ring spectra \(TC(*,p)\to TC^{(1)}(*,p)\to \mathbb{S}^{C_{p}}\). The main result of the present paper is a calculation of the Dyer-Lashof operations \(Q^i\) in \(H_*(TC^{(1)}(*,2))\) and \(H_*(\mathbb{S}^{C_2})\), where \(H_*(TC^{(1)}(*,2))\) is mod 2 spectrum homology. As a corollary, the Dyer-Lashof operations \(Q^{2i}\) for \(TC(*,2)\) follow and in particular, they are non-trivial. \(Q^{2i+1}\) are all zero. Moreover, the \(G\)-fixed point of the sphere spectrum \(\mathbb{S}^G\) and the algebraic \(K\)-theory spectrum of the category of finite \(G\)-sets are compared as structured spectra
reviewed by: Lisbeth Fajstrup
Mathematics Subject Classification ID: 55S12
Mathematics Subject Classification ID: 55P43
Mathematics Subject Classification ID: 19D10
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 55P92
zbMATH DE Number: 5673942
zbMATH Keywords: topological cyclic homology
zbMATH Keywords: homology operation
zbMATH Keywords: algebraic \(K\)-theory
zbMATH Keywords: Dyer Lashof operations
zbMATH Keywords: structured ring spectra
MaRDI profile type: Publication
full work available at URL: gt.2010.14.755
OpenAlex ID: W2093441604
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DOI: 10.2140/GT.2010.14.755
description: scientific article; zbMATH DE number 5673942