Koszul duality complexes for the cohomology of iterated loop spaces of spheres

TL;DR

This paper constructs an explicit algebraic complex to compute the cohomology of p-profinite completions of iterated loop spaces of spheres, using Koszul duality and E_n-operads.

Contribution

It introduces a new algebraic complex based on E_n-operads and Koszul duality to explicitly compute cohomology of iterated loop spaces of spheres.

Findings

Explicit algebraic complex for cohomology computation

Connection between loop space cohomology and E_n-operad algebras

Koszul duality for E_n-operads established

Abstract

The goal of this article is to make explicit a structured complex whose homology computes the cohomology of the p-profinite completion of the n-fold loop space of a sphere of dimension d=n-m<n. This complex is defined purely algebraically, in terms of characteristic structures of E_n-operads. Our construction involves: the free complete algebra in one variable associated to any E_n-operad; and an element in this free complete algebra, which is associated to a morphism from the operad of L-infinity algebras to an operadic suspension of our E_n-operad. We deduce our main theorem from: a connection between the cohomology of iterated loop spaces and the cohomology of algebras over E_n-operads; and a Koszul duality result for E_n-operads.