Adams operations, algebras up to homotopy and cyclic homology (Q1585056)

A new \(A_{\infty}\)-algebra structure on the cyclic bar complex of a commutative algebra over the rationals is defined. In contrast with the \(A_{\infty}\)-structure of \textit{E. Getzler} and \textit{J. D. S. Jones} [Ill. J. Math. 34, No. 2, 256-283 (1990; Zbl 0701.55009)], it is shown that the Adams operations are \(A_{\infty}\)-algebra endomorphisms for this new structure. The methods rely on geometrical constructions with convex polytopes, extending the techniques developed by the author for Hochschild homology [\textit{F. Patras}, Bull. Soc. Math. Fr. 119, No. 2, 173-198 (1991; Zbl 0752.55014)]. The maps of the new \(A_{\infty}\)-structure are associated to polytopes constructed from standard simplexes using a twisted cartesian product. The resulting \(A_{\infty}\)-structure on the cyclic bar complex induces the Loday-Quillen product in cyclic homology. Similarly, \(A_{\infty}\)-algebra structures are obtained for the negative and periodic cyclic complexes and these induce the Hood-Jones product in homology.