On the Eilenberg-Zilber theorem for crossed complexes
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)].
scientific article; zbMATH DE number 1895974
scientific article