Koszul duality for topological En‐operads. Issue 1 (10th May 2022)

• Record Type: Journal Article
• Authors: Ching, Michael; Salvatore, Paolo
• Is Part Of: Proceedings of the London Mathematical Society. Volume 125:Issue 1(2022)
• Page Start: 1
• Page End: 60
• Publication Date: 2022-05-10
• Subjects: Mathematics -- Periodicals; Mathematics; Periodicals; 510
• DOI: 10.1112/plms.12442
• Languages: English
• ISSNs: 0024-6115

Abstract

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.