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. The purpose of these notes is to explain the relationship between Batanin’s categories of pruned trees and iterated bar complexes. This article is an appendix of the article [4]. Our purpose is to explain the relationship between Batanin’s categories of pruned trees (see [1, 2]) and iterated bar complexes and to revisit some constructions [4] in this formalism. The reader can use this appendix as an informal introduction to the constructions of [4]. These notes is the appendix part of a preliminary version of [4], extracted without changes from this article except that we have removed the appendix mark from paragraph numberings. Thus the reader can easily retrieve references to former versions of [4] in this manuscript. 1. Level trees and sequences of non-decreasing surjections. Our first aim is to make explicit the expansion of iterated tensor coalgebras (T c Σ) n (M), for a connected Σ∗module

The standard reduced bar complex B(A) of a differential graded algebra A inherits a natural commutative algebra structure if A is a commutative algebra. We address an extension of this construction in the context of E-infinity algebras. We prove that the bar complex of any E-infinity algebra can be equipped with the structure of an E-infinity algebra so that the bar construction defines a functor from E-infinity algebras to E-infinity algebras. We prove the homotopy uniqueness of such natural E-infinity structures on the bar construction. We apply our construction to cochain complexes of topological spaces, which are instances of E-infinity algebras. We prove that the n-th iterated bar complexes of the cochain algebra of a space X is equivalent to the cochain complex of the n-fold iterated loop space of X, under reasonable connectedness, completeness and finiteness assumptions on X. Key words: Bar construction, E-infinity algebras, iterated loop spaces 2000 MSC: 57T30, 55P48, 18G55, 55P35

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