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

We study a cofibrant E-infty 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.