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An explicit formula for the generalized cyclic shuffle map. (English) Zbl 1286.19005

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 \(X\).

MSC:

19D55 \(K\)-theory and homology; cyclic homology and cohomology
05E45 Combinatorial aspects of simplicial complexes
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