Formality conjectures and deformation

Let M be a Poisson manifold. Kontsevich proved that star products exist on M and he gave a classification. To relate his classification with other cla ssifications, one could try to extend the Connes-Flato-Sternheimer invarian t to a general Poisson manifold. We show how generalization of this invaria nt is related to the formality conjecture for chains. Finally, we show how to prove those conjectures 'step by step'. Our approach, different from Tam arkin's, will give explicit formulas but doesn't yet solve the general conj ecture.