On 6-connected finite H-spaces with two torsion
type: Item
instance of: scholarly article
title: On 6-connected finite H-spaces with two torsion
zbMATH Open document ID: 0681.55008
DOI: 10.1016/0040-9383(89)90029-3
author: James P. Lin
author: Frank Williams
published in: Topology
publication date: 1989-01-01T00:00:00Z
review text: When realizing a Hopf algebra over the mod 2 Steenrod Algebra as the cohomology of an H-space, there are many restrictions. Nevertheless, for several Hopf algebras it is unknown if they can be realized as the cohomology of a finite mod 2 H-space. The Hopf algebra \(A=({\mathbb{Z}}_ 2[x_ 7]/(x_ 7)^ 4)\otimes \wedge (x_{11},x_{13})\) is perhaps the most natural candidate to be realized as the cohomology of a finite H-space. This work shows that there is no H-space X such that \(H^*(X,{\mathbb{Z}}_ 2)\approx A\). The authors study the cohomology of \(\Omega\) X and use that \(\Omega\) X is a homotopy commutative H-space. A tertiary cohomology operation is used in order to detect a class in \(H^*(\Omega X,{\mathbb{Z}}_ 2)\) which leads to a contradiction for the existence of the H-space X. This work reinforces the idea that there is no new mod 2 finite H-space. The known examples of mod 2 finite H-spaces are just a product of seven spheres by compact Lie groups.
Mathematics Subject Classification ID: 55P45
Mathematics Subject Classification ID: 55S20
Mathematics Subject Classification ID: 57T25
Mathematics Subject Classification ID: 57T05
zbMATH DE Number: 4115529
zbMATH Keywords: realizing a Hopf algebra over the mod 2 Steenrod Algebra
zbMATH Keywords: cohomology of an H-space
zbMATH Keywords: tertiary cohomology operation
zbMATH Keywords: mod 2 finite H-space
reviewed by: Daciberg Lima Gonçalves
MaRDI profile type: Publication
full work available at URL: 0040-9383(89)90029-3
OpenAlex ID: W2070588844
description: scientific article; zbMATH DE number 4115529