Koszul duality for topological En‐operads

We show that the Koszul dual of an En$E_n$‐operad in spectra is O(n)$O(n)$‐equivariantly equivalent to its n$n$‐fold desuspension. To this purpose we introduce a new O(n)$O(n)$‐operad of Euclidean spaces Rn$R_n$, the barycentric operad, that is fibred over simplexes and has homeomorphisms as structure maps; we also introduce its suboperad of restricted little n$n$‐discs Dn$D_n$, that is an En$E_n$‐operad. The duality is realised by an unstable explicit S‐duality pairing (Fn)+∧BDn→S¯n$(F_n)_+ \wedge BD_n \rightarrow \bar{S}_n$, where B$B$ is the bar‐cooperad construction, Fn$F_n$ is the Fulton–MacPherson En$E_n$‐operad, and the dualising object S¯n$\bar{S}_n$ is an operad of spheres that are one‐point compactifications of star‐shaped neighbourhoods in Rn$R_n$. We also identify the Koszul dual of the operad inclusion map En→En+m$E_n \rightarrow E_{n+m}$ as the (n+m)$(n+m)$‐fold desuspension of an unstable operad map En+m→ΣmEn$E_{n+m} \rightarrow \Sigma ^m E_n$ defined by May.