Koszul duality for topological E_n-operads

Abstract

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 ) + ∧BD n → Sn , where B is the bar-cooperad construction, F n is the Fulton-MacPherson E n -operad, and the dualizing object Sn 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 → E n+m as the (n + m)-fold desuspension of an unstable operad map E n+m → Σ m E n defined by May.

Partition complexes and trees

“…The reader might compare this to the remark immediately following [3,Lemma 9.5]. This reference contains a much more detailed analysis of the bar construction of the Fulton-MacPherson operad and includes a proof of the 'self-duality' of the E d -operad.…”