The Morava \(K\)-theory of spaces related to \(BO\)
type: Item
instance of: scholarly article
title: The Morava \(K\)-theory of spaces related to \(BO\)
zbMATH Open document ID: 1063.55002
author: Nitu Kitchloo
author: Gerd Laures
author: W. Stephen Wilson
published in: Advances in Mathematics
publication date: 2004-11-26T00:00:00Z
review text: Let \({bo}_*\) denote the \(\Omega\)-spectrum with \({bo}_0 = {\mathbb Z} \times BO\) and \({BG}_*\) denote the \(\Omega\)-spectrum with \({BG}_0 = BG\) for \(G = O\), \(SO\) or \(Spin\). This paper computes the Morava \(K\)-theory at the prime \(2\), denoted \(K_*(\enskip)\), of the spaces in these \(\Omega\) spectra. These spaces are related by a sequence of connective covering maps and so these computations can be done inductively by studying such coverings. The authors show that if \(B = {bo}_i\) with \(i \geq 4\) or \(B= {BG}_i\) with \(i \geq 2\) and \(E \to B\) is a connective cover with fibre \(F\) then there is a short exact sequence of Hopf algebras \(K_* \to K_* F \to K_* E \to K_* B \to K_*\). The authors also describe what happens when \(B = {bo}_i\) with \(i < 4\) or \(B = BG_i\) with \(i < 2\) (the situation here is sufficiently varied and complicated that the statement of results takes up more than 3 pages of the paper). Of particular interest is their description of the Morava \(K\)-theory of the map \({bo}_8 \to {BSpin}_0\) which fits into a 6-term exact sequence of Hopf algebras. Results corresponding to these for \(BU\) were previously established by \textit{D. C. Ravenel, W. S. Wilson} and \textit{N. Yagita} [K-theory 15, 149-199 (1998; Zbl 0912.55002)]. In the present paper the authors identify the Hopf algebra cokernel of the forgetful map \(BU \to BO\) and show that it is isomorphic to \(K_* (\Pi_{i >0} K({\mathbb F}_2,i))\). They also use their results to compute the 2-adicly completed Brown-Peterson cohomology of many of the spaces in these \(\Omega\)-spectra.
reviewed by: Keith Johnson
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55N20
zbMATH DE Number: 2119097
zbMATH Keywords: \(K\)-theory
zbMATH Keywords: Morava \(K\)-theory
zbMATH Keywords: Brown-Peterson Cohomology
MaRDI profile type: Publication
full work available at URL: j.aim.2003.10.008
OpenAlex ID: W2034163506
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DOI: 10.1016/J.AIM.2003.10.008
description: scientific article; zbMATH DE number 2119097