mod 3 cohomology algebras of finite \(H\)-spaces
type: Item
instance of: scholarly article
title: mod 3 cohomology algebras of finite \(H\)-spaces
zbMATH Open document ID: 1004.55014
author: James P. Lin
published in: Mathematische Zeitschrift
publication date: 2002-09-17T00:00:00Z
review text: Let \(X\) be a \(3\)-local, finite, simply connected \(H\)-space whose homology is associative. For example, \(X\) could be a localized odd dimensional sphere, the Lie group \(E_{8}\) or the \(H\)-space \(X(3)\) constructed by \textit{J. R. Harper} [\(H\)-spaces with torsion, Mem. Am. Math. Soc. 223 (1979; Zbl 0421.55006)]. The main theorem of this paper states that, up to isomorphism of mod \(3\) cohomology algebras, these examples, and products of them, are the only ones. Concretely, the mod \(3\) cohomology of \(X\) must be isomorphic, as an algebra, to the cohomology of a product of copies of \(X(3)\), copies of \(E_{8}\) and odd dimensional spheres. There are also partial results to suggest that an isomorphism as algebras over the Steenrod algebra may also be possible, as demonstrated by \textit{R. Kane} [J. Pure Appl. Algebra 41, 213-232 (1986; Zbl 0635.57027)] in the case where there is exactly one copy of \(E_{8}\).
Mathematics Subject Classification ID: 55S20
Mathematics Subject Classification ID: 55P45
Mathematics Subject Classification ID: 55R35
Mathematics Subject Classification ID: 55S45
Mathematics Subject Classification ID: 57T25
zbMATH DE Number: 1801564
zbMATH Keywords: \(H\)-spaces
zbMATH Keywords: cohomology algebra
reviewed by: Martin D. Crossley
MaRDI profile type: Publication
full work available at URL: s002090100389
OpenAlex ID: W1989152079
DOI: 10.1007/S002090100389
description: scientific article; zbMATH DE number 1801564