Th\'eor\`eme d'Eilenberg-Zilber en homologie cyclique enti\`ere

Abstract

For simplicial modules, Eilenberg-Zilber's classical theorem states the existence of a product $sh : M\otimes N\to M\times N$ (the shuffle) and a coproduct $AW : M\times N\to M\otimes N$ (the Alexander-Whitney map), which are quasi-inverse of eachother. A cyclic version of this theorem was established in 1987 by Hood and Jones: they proved that $sh$ and $AW$ admit "coextensions" $sh_\infty$ and $AW_\infty$, using an acyclic-model method. Besides, an explicit formula for $sh_\infty$ has been discovered by several authors. But the question remained open of such an explicit formula for $AW_\infty$, and for the homotopies by which $sh_\infty$ and $AW_\infty$ are mutual quasi-inverses and are quasi-(co)-associative. We present a complete answer to this problem and show that all these -- now explicit -- maps extend continuously to entire cyclic complexes (associated to normed algebras).