Koszul duality for topological E n ‐operads

Abstract

We show that the Koszul dual of an E n $E_n$ -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 R n $R_n$ , 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 D n $D_n$ , that is an E n $E_n$ -operad. The duality is realised by an unstable explicit S-duality pairing ( F n ) + ∧ B D n → S ¯ n $(F_n)_+ \wedge BD_n \rightarrow \bar{S}_n$ , where B $B$ is the bar-cooperad construction, F n $F_n$ is the Fulton–MacPherson E n $E_n$ -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 R n $R_n$ . We also identify the Koszul dual of the operad inclusion map E n → E n + m $E_n \rightarrow E_{n+m}$ as the ( n + m ) $(n+m)$ -fold desuspension of an unstable operad map E n + m → Σ m E n $E_{n+m} \rightarrow \Sigma ^m E_n$ defined by May.