2016/11/25 by Anne Bauval, Bauval, Anne · 7 citations
Mathematics · #16E40 #19D55 #Advanced Topics in Algebra #Algebra over a field #Algebraic structures and combinatorial models #Associative property #Cohomology #Combinatorics #Coproduct #Cyclic homology #FOS: Mathematics #Geometry #Homotopy and Cohomology in Algebraic Topology #Inverse #K-Theory and Homology (math.KT) #Mathematics #Product (mathematics) #Pure mathematics #math.KT #msc:16E40 #msc:19D55
paper · pdf · doi:10.48550/arxiv.1611.08437
published in arXiv (Cornell University) (Cornell University) · in French, Preprint written and disseminated in 1998
arxiv created 2016/11/25 · openalex publication_date 2016/11/25 · arxiv updated 2016/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For simplicial modules, Eilenberg-Zilber's classical theorem states the existence of a product sh : M⊗ N→ M× N (the shuffle) and a coproduct AW : M× N→ M⊗ 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_∞ and AW_∞, using an acyclic-model method. Besides, an explicit formula for sh_∞ has been discovered by several authors. But the question remained open of such an explicit formula for AW_∞, and for the homotopies by which sh_∞ and AW_∞ 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).