Zhang, Jiao; Wang, Qing-Wen An explicit formula for the generalized cyclic shuffle map. (English) Zbl 1286.19005 Can. Math. Bull. 57, No. 1, 210-223 (2014). The shuffle product of cyclic manifolds is used in the definition of products and coproducts in cyclic (co)homology. Also, it is used for proving homological properties of cyclic (co)homology, like the Eilenberg-Zilber Theorem and Künneth type formulas.The authors, in the paper under review, define the generalized shuffle product for any cylindrical module. Earlier, the shuffle produce was define on cylindrical modules that could decompose into a product of cyclic modules. The definition uses partitions. Furthermore, they prove certain combinatorial properties for their construction. They use their calculations to show an Eilenberg-Zilber type theorem for cylindrical modules. More specifically, let \(\zeta\) the cyclic shuffle product and \(\xi\) the generalized shuffle product. For a cylindrical module \(X\), they prove that the map \(({\zeta}, {\xi})\) induces an isomorphism of cyclic homologies \(HC_*(\text{Tot}(X)) \cong HC_*({\Delta}(X))\), where \(\text{Tot}(X)\) (respectively \({\Delta}(X)\)) is the total (respectively diagonal) complex of \(X\). Reviewer: Stratos Prassidis (Karlovassi) Cited in 2 Documents MSC: 19D55 \(K\)-theory and homology; cyclic homology and cohomology 05E45 Combinatorial aspects of simplicial complexes Keywords:generalized cyclic shuffle map; cylindrical module; Eilenberg-zilber theorem; cyclic homology generalized cyclic shuffle map; cylindrical module; Eilenberg-zilber theorem; cyclic homology PDFBibTeX XMLCite \textit{J. Zhang} and \textit{Q.-W. Wang}, Can. Math. Bull. 57, No. 1, 210--223 (2014; Zbl 1286.19005) Full Text: DOI