On the Eilenberg-Zilber theorem for crossed complexes
type: Item
instance of: scholarly article
title: On the Eilenberg-Zilber theorem for crossed complexes
zbMATH Open document ID: 1038.55012
DOI: 10.1016/S0022-4049(02)00160-3
author: A. P. Tonks
published in: Journal of Pure and Applied Algebra
publication date: 2003-04-09T00:00:00Z
review text: The original Eilenberg-Zilber theorem states that if \(X\) and \(Y\) are simplicial sets then there exists a chain homotopy equivalence \[ C_N(X)\otimes C_N(Y)\simeq C_N(X\times Y), \] where \(C_N(X)\) is the normalized free chain complex on the simplicial set \(X\). This result was generalized in different contexts by Dold and Puppe, Kassel, Loday and others. In the present paper, which is based on his thesis, the author generalizes the Eilenberg-Zilber theorem by considering the appropriate tensor product of crossed complexes. For this purpose, he proves that the crossed complex \(\pi(X)\otimes \pi(Y)\) is a natural strong deformation rectract of \(\pi(X\times Y)\), where \(\pi(-)\) is the fundamental crossed complex functor that associates a crossed complex to each simplicial set. More precisely, there are maps \(a:\pi(X)\otimes \pi(Y)\rightarrow \pi(X\times Y)\) and a one-sided inverse \(b\) such that \(ba\) is homotopic to the identity. Certain associativity and interchange relations for this natural transformation are also proved. Applications to equivariant topology appear in \textit{R. Brown, M. Golasinski, T. Porter} and \textit{A. Tonks} [K-Theory 23, No. 2, 129--155 (2001; Zbl 1012.55018)].
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 18G50
zbMATH DE Number: 1895974
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1895974