Koszul Duality Complexes for the Cohomology of Iterated Loop Spaces of Spheres

— The goal of this article is to make explicit a structured complex computing the cohomology of a profinite completion of the n-fold loop space of a sphere of dimension d < n. Our construction involves: the free complete algebra in one variable associated to any fixed En-operad; an element in this free complete algebra determined by a morphism from the operad of L∞-algebras to an operadic suspension of our En-operad. As such, our complex is defined purely algebraically in terms of characteristic structures of En-operads. We deduce our main theorem from several results obtained in a previous series of articles – namely: a connection between the cohomology of iterated loop spaces and the cohomology of algebras over En-operads, and a Koszul duality result for En-operads. We use notably that the Koszul duality of En-operads makes appear structure maps of the cochain algebra of spheres.