Compatibility with cap-products in Tsygan's formality and homological Duflo isomorphism
Résumé
In this paper we prove that the isomorphism induced on tangent homology by the Shoikhet-Tsygan formality isomorphism for Hochschild chains is compatible with cap-products. This is a homological analog of the compatibility with cup-products of the isomorphism induced on tangent cohomology by Kontsevich formality isomorphism for Hochschild cochains. As in the cohomological situation our proof relies on a homotopy argument. This homotopy argument involves a 2-dimensional surface in a 3-dimensional configuration space, the {\bf I-cube} (which plays the rôle of {\bf Kontsevich's eye}). As an application, we state and prove a homological version of the Duflo isomorphism.