Sur l'itération de la construction Cobar

Abstract

Since Serre's work [ 12], we know that loop spaces play a central role in algebraic topology. In particular, iterated loop spaces pose the tough problem which consists in iterating the Cobar construction (see [ 2], [ 10], [ 11], [ 14] and [ 15]). In this Note, to make progress in this topic, we give explanations and complements about Adams ' relation [ 1] between C *( ΩX) and Cobar C *(X) (Z, Z) when X is the suspension of a reduced (with trivial 0-skeleton) simplicial set, Ω being here the simplicial Kan model [ 8] of the loop space functor. In particular, our results give a surprising experimental fact: the existence of an “exotic” differential which can replace the classical Adams differential in the Cobar construction. They also permit us to obtain with a new method some previous results of Baues ([ 2], [ 3]) about Ω 2 X when X is the suspension of a 1-reduced (with trivial 1-skeleton) simplicial set.