Two-primary algebraic \(K\)-theory of pointed spaces
type: Item
instance of: scholarly article
title: Two-primary algebraic \(K\)-theory of pointed spaces
zbMATH Open document ID: 1009.19001
DOI: 10.1016/S0040-9383(01)00005-2
author: John Rognes
published in: Topology
publication date: 2002-09-17T00:00:00Z
review text: In this impressive paper the author computes the \(2\)-primary homotopy type of Waldhausen's algebraic K-theory spectrum of a point \(A(*)\). This has far-reaching consequences; for instance, \(A(*)\) splits into two pieces, one of which is the sphere spectrum and the other is the smooth Whitehead spectrum \(\text{Wh}^{\text{Diff}}(*)\). The Whitehead spectrum in turn gives information about the group of diffeomorphisms of high-dimensional discs. To get optimal results in this direction, one needs to understand the involutive refinement of the results of the paper under review, and the author announces calculations to come in a later paper. The mathematics used in this paper is highly varied. Key ingredients are 1) Voevodsky's proof of the Milnor conjecture, and the proof of the Lichtenbaum-Quillen conjecture for the integers at two due to \textit{J. Rognes} and \textit{C. Weibel} [J. Am. Math. Soc. 13, No. 1, 1-54 (2000; Zbl 0934.19001)]; 2) the author's calculation [\textit{J. Rognes}, J. Pure Appl. Algebra 134, No. 3, 287-326 (1999; Zbl 0929.19004)] of topological cyclic homology of the integers at \(2\); 3) \textit{M. Bökstedt, W. C. Hsiang} and \textit{I. Madsen}'s analysis of the topological cyclic homology of spaces [Invent. Math. 111, No. 3, 465-539 (1993; Zbl 0804.55004)] (the author provides a missing part in the argument), and the reviewer's result [\textit{B. I. Dundas}, Acta Math. 179, No. 2, 223-242 (1997; Zbl 0913.19002)] relating K-theory and topological cyclic homology through linearization; 4) an analysis of the mod \(2\) cohomology of the Thom spectrum \({\mathbb C}P^{\infty}_{-1}\) of minus the canonical line bundle over \({\mathbb C}P^{\infty}\). In what follows all spectra will tacitly be completed at \(2\). The calculation is performed by computing the mod \(2\) cohomology of \(A(*)\) as a module over the mod \(2\) Steenrod algebra. This is done by using the results in point 3) to convert the problem into a problem in topological cyclic homolo…
Mathematics Subject Classification ID: 19D10
Mathematics Subject Classification ID: 55N10
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 55Q10
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55T15
Mathematics Subject Classification ID: 57Q60
Mathematics Subject Classification ID: 57R50
zbMATH DE Number: 1801882
zbMATH Keywords: algebraic K-theory of spaces
zbMATH Keywords: stable smooth concordance space
zbMATH Keywords: topological cyclic homology
zbMATH Keywords: cyclotomic trace map
zbMATH Keywords: truncated complex projective space
zbMATH Keywords: spectrum cohomology
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: Adams spectral sequence
zbMATH Keywords: linearization map
zbMATH Keywords: Hatcher-Waldhausen map
zbMATH Keywords: rigid tube map
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1801882