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)_+ BD_n → 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 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.