Towards the algebra structure of the Morava \(K\)-theory of the orthogonal groups
type: Item
instance of: scholarly article
title: Towards the algebra structure of the Morava \(K\)-theory of the orthogonal groups
zbMATH Open document ID: 0899.55005
DOI: 10.1007/BF02677854
published in: Manuscripta Mathematica
publication date: 1998-11-11T00:00:00Z
full work available at URL: https://eudml.org/doc/156339
review text: It is well known that \(BP_*= \mathbb{Z}_{(2)} [v_1,v_2, \dots]\) where the degree of \(v_i\) is \(2(2^i-1)\). Using the Sullivan-Baas technique, Jack Morava kills \(\{v_i \mid i<\ell\}\), where \(\ell>0\), to get \(P(\ell)\) a \(BP\)-module theory with coefficient ring \(P(\ell)_* =\mathbb{Z}/2 [v_\ell, v_{\ell+1}, \dots]\). In this note, the author gives the generators for \(K(\ell)\)-homology of Lie groups \(\text{SO} (2^{\ell+1}-1)\) and \(\text{Spin} (2^{\ell +1}-1)\) -- see Proposition 2.4. and section 3. In section 5, he also determines a complete set of algebra relations for the generators -- i.e. the algebra structure -- of \(\text{Pl}_* \text{SO} (2^{\ell +1}-1)\) and \(\text{Pl}_* \text{Spin} (2^{\ell +1}_1-1)\) respectively. The formulas look too complicated to be reproduced here. The results of this paper can also be used to give another derivation of the Hopf algebra structure of \(H^* (\text{Spin} (n); \mathbb{Z}_2)\) -- see Corollary 5.6. combined with Proposition 2.1.
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 57T10
Mathematics Subject Classification ID: 55N99
zbMATH DE Number: 1135240
zbMATH Keywords: Morava \(K\)-theory
zbMATH Keywords: Hopf algebra
author: Vidhyanath K. Rao
reviewed by: Cristian N. Costinescu
MaRDI profile type: Publication
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OpenAlex ID: W2065244401
description: scientific article; zbMATH DE number 1135240