The Hopf ring for \(bo\) and its connective covers
type: Item
instance of: scholarly article
title: The Hopf ring for \(bo\) and its connective covers
zbMATH Open document ID: 1115.55001
author: Dena S. Cowen Morton
published in: Journal of Pure and Applied Algebra
publication date: 2007-04-26T00:00:00Z
review text: If \(F_*(\enskip)\) is a multiplicative homology theory and \(E_*\) is a ring spectrum then the homology groups \(F_*(E_*)\) have a coproduct, conjugation, counit and a pair of products which give them the structure of a Hopf ring, i.e., a ring object in the category of coalgebras. The study of such objects was initiated by \textit{D. C. Ravenel} and \textit{W. S. Wilson} [J. Pure Appl. Algebra 9, 241--280 (1977; Zbl 0373.57020)] and their computation is of interest to topologists because of their connection with the unstable operations between \(E^* (\enskip)\) and \(F^*(\enskip)\). The paper under review gives a complete computation of this Hopf ring for \(E_* = bo\), the connective real \(K\)-theory spectrum, and \(F_* = \overline{H}_* = H_*(\enskip ; {\mathbb Z}/2)\), ordinary mod \(2\) homology. In an earlier paper the author and \textit{N. Strickland} [J. Pure Appl. Algebra 166, 247--265 (2002; Zbl 0987.55007)] computed the corresponding Hopf ring for nonconnective real \(K\)-theory, \(\overline{H}_*(KO_*)\). The case of integral ordinary homology, \(H_*(\overline{H}_*)\), is classical. The author shows that \(\overline{H}_*(bo_*)\) injects into the product \(H_* (KO) \otimes H_* (\overline{H}_*)\) and she describes the image. The result is a very large graded tensor product of families of polynomial and exterior algebras coming from these two factors. The method of computation is repeated use of the bar spectral sequence. The paper concludes with the computation of \(\overline{H}_*(bo \langle n \rangle)\), the homology of the connective covers of \(bo\), as Hopf modules over \(\overline{H}_* (bo_*)\).
reviewed by: Keith Johnson
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 55P43
zbMATH DE Number: 5146871
zbMATH Keywords: Hopf ring
zbMATH Keywords: bar spectral sequence
zbMATH Keywords: real connective \(K\)-theory
MaRDI profile type: Publication
full work available at URL: j.jpaa.2006.09.014
OpenAlex ID: W146685433
cites work: The Hopf rings for connective Morava K-theory and connective complex K- theory
cites work: Q3697897
cites work: The Hopf rings for \(KO\) and \(KU\)
cites work: The Hopf ring for complex cobordism
cites work: The Morava K-Theories of Eilenberg-MacLane Spaces and the Conner-Floyd Conjecture
cites work: Determination of H ∗ (BO(k,...,∞), Z 2 ) and H ∗ (BU(k,..., ∞),Z 2 )
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cites work: Hopf rings in algebraic topology
DOI: 10.1016/J.JPAA.2006.09.014
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description: scientific article; zbMATH DE number 5146871