We review definitions and basic properties of operads, PROPs and algebras over these structures.

Abstract. 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. Contents

Abstract. 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. Contents