Steenrod operations in \(shc\)-algebras
type: Item
instance of: scholarly article
title: Steenrod operations in \(shc\)-algebras
zbMATH Open document ID: 1052.55008
author: Bitjong Ndombol
author: Jean-Claude Thomas
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: Let \(\pi\) be a cyclic group of order a fixed prime \(p\) and \(W\) be the standard free resolution of \(\mathbb{F}_p\) as a \(\mathbb{F}_p[\pi]\)-module. Generalizing May's work, a differential graded algebra \(A\) is called a \(\pi\)-\(shc\)-algebra if \(A\) is an \(shc\) algebra with a morphism of differential graded \(\pi\)-algebras \[ W\otimes\Omega B (A^{\otimes p})\to A, \] whose restriction to \(\Omega\otimes \Omega B (A^{\otimes p})\) consists into the \(shc\)-multiplication \(\Omega B(A^{\otimes p})\to \Omega B A\) followed by the augmentation. An example is given by the singular cochains \(N^*X\) of a 1-connected space with coefficients in \(\mathbb{F}_p\). The authors prove that if \(A\) is a \(\pi\)-\(shc\)-algebra, then the Hochschild homology \(HH_*A\) admits Steenrod operations. They also show that the Jones isomorphism \[ HH_* N^*X\cong H^* (X^{S^1}; \mathbb{F}_p) \] preserves Steenrod operations.
Mathematics Subject Classification ID: 55P35
Mathematics Subject Classification ID: 13D03
Mathematics Subject Classification ID: 55P48
Mathematics Subject Classification ID: 16E45
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 2093431
zbMATH Keywords: Steenrod operations
zbMATH Keywords: free loop space homology
reviewed by: Yves Félix
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.02.003
OpenAlex ID: W2047932396
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DOI: 10.1016/J.JPAA.2004.02.003
description: scientific article; zbMATH DE number 2093431