An E∞-extension of the associahedra and the Tamarkin cell mystery

In this note based on the author's communication with M. Batanin, we study a cofibrant E∞-operad generated by the Fox-Neuwirth cells of the configuration space of points in the Euclidean space. We show that, below the 'critical dimensions' in which 'bad cells' exist, this operad is modeled by the geometry of the Fulton-MacPherson compactification of this configuration space. We analyze the Tamarkin bad cell and calculate the differential of the corresponding generator. We also describe a simpler, four-dimensional bad cell. We finish the paper by proving an auxiliary result giving a characterization, over integers, of free Lie algebras.