Koszul duality for topological En‐operads. Issue 1 (10th May 2022) - Ching, Michael; Salvatore, Paolo
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 $(F_n)_+ \wedge BD_n → \bar{S}_n$, 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 $E_n → E_{n+m}$ as the ( n + m ) $(n+m)$ ‐fold desuspension of an unstable operad map E n + m → Σ m E n $E_{n+m} → Σ ^m E_n$ defined by May.