Paper detail

Koszul duality for topological E_n-operads

We show that the Koszul dual of an E_n-operad in spectra is O(n)-equivariantly equivalent to its n-fold desuspension. To this purpose we introduce a new O(n)-operad of Euclidean spaces R_n, the barycentric operad, that is fibred over simplexes and has homeomorphisms as structure maps; we also introduce its sub-operad of restricted little n-discs D_n, that is an E_n-operad. The duality is realized by an unstable explicit S-duality pairing (F_n)_+ \smash BD_n \to S_n, where B is the bar-cooperad construction, F_n is the Fulton-MacPherson E_n-operad, and the dualizing object S_n is an operad of spheres that are one-point compactifications of star-shaped neighbourhoods in R_n. We also identify the Koszul dual of the operad inclusion map E_n \to E_{n+m} as the (n+m)-fold desuspension of an unstable operad map E_{n+m} \to Σ^m E_n defined by May.