The homology algebra of finite H-spaces
type: Item
instance of: scholarly article
title: The homology algebra of finite H-spaces
zbMATH Open document ID: 0635.57027
DOI: 10.1016/0022-4049(86)90110-6
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: Let \(\mu\) : \(X\times X\to X\) be the multiplication of an H-space X, H \(*=H\) *(X,\({\mathbb{Z}}/(p))\), \(H_*=H_*(X,{\mathbb{Z}}/(p))\). The prime p is assumed to be odd, X simply connected, \(H_*\) finite with associative Pontrjagin product. Finally, it is assumed that x \(p=0\) for all x in the augmentation ideal \(\tilde H\) *. The paper studies the implications of the action of the Steenrod algebra \(A=A\) *(p) of all stable cohomology operations with \({\mathbb{Z}}/(p)\) coefficients on the module of indecomposables QH * on the Lie algebra structure of \(PH_*\), the primitive elements in \(H_*\). Here is a sample result (Theorem 1.2): Suppose \(p=3\). a) If rank X\(<8\), then A acts trivially on \(Q^{even}H\) *. b) If rank X\(=8\) and A acts non-trivially on \(Q^{even}H\) *, H *\(\approx H\) \(*(E_ 8,{\mathbb{Z}}/(3))\) as Hopf algebras over A.
Mathematics Subject Classification ID: 57T25
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 57T05
Mathematics Subject Classification ID: 55P45
zbMATH DE Number: 4034689
zbMATH Keywords: finite H-space
zbMATH Keywords: primitive elements of the homology coalgebra
zbMATH Keywords: exceptional Lie group \(E_ 8\)
zbMATH Keywords: indecomposables of the cohomology ring
zbMATH Keywords: associative Pontrjagin product
zbMATH Keywords: action of the Steenrod algebra
zbMATH Keywords: rank
author: Richard Kane
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90110-6
OpenAlex ID: W2057754699
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description: scientific article; zbMATH DE number 4034689