Koszul duality for topological E _n ‐operads

Abstract

We show that the Koszul dual of an 𝐸 𝑛 -operad in spectra is 𝑂(𝑛)-equivariantly equivalent to its 𝑛-fold desuspension. To this purpose we introduce a new 𝑂(𝑛)operad of Euclidean spaces 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-discs 𝐷 𝑛 , that is an 𝐸 𝑛 -operad. The duality is realised by an unstable explicit S-duality pairing (𝐹 𝑛 ) + ∧ 𝐵𝐷 𝑛 → S𝑛 , where 𝐵 is the bar-cooperad construction, 𝐹 𝑛 is the Fulton-MacPherson 𝐸 𝑛 -operad, and the dualising object S𝑛 is an operad of spheres that are one-point compactifications of star-shaped neighbourhoods in 𝑅 𝑛 . We also identify the Koszul dual of the operad inclusion map 𝐸 𝑛 → 𝐸 𝑛+𝑚 as the (𝑛 + 𝑚)-fold desuspension of an unstable operad map 𝐸 𝑛+𝑚 → Σ 𝑚 𝐸 𝑛 defined by May.

Koszul self‐duality of manifolds

“…Let us now define the Koszul dual cooperad. This version appears in [17,Section 6] where it is observed that it coincides with the version originally defined in [15]. It bears the name "bar construction" because Ching observed in his thesis that it is isomorphic to 𝐵(1, 𝑂, 1), a bar construction with respect to the •-product.…”

“…In this section, we will show how the existence of a Koszul zigzag for $$\Sigma ^\infty _+\mathcal {F}_n$$ gives rise to a Koszul zigzag for $$(\mathcal {F}_M,\widehat{\partial }\mathcal {F}_M)$$ when $$M$$ is a compact, codimension 0 submanifold of $$\mathbb {R}^n$$. It is known that such Koszul zigzags exist for $$\Sigma ^\infty _+ \mathcal {F}_n$$ [17, Theorem 1.1]: Theorem There is a zigzag of equivalences of operads from $$\Sigma ^\infty _+ \mathcal {F}_n$$ to the $$n$$‐fold suspension of its Koszul dual: $$$$\begin{equation*} \Sigma ^\infty _+ \mathcal {F}_n \simeq \dots \simeq s_n K(\Sigma ^\infty _+ \mathcal {F}_n). \end{equation*}$$$$…”

“…After much speculation, this was recently realized by Ching–Salvatore [17]. To complete the program of translating Poincaré–Koszul duality into a more classical form, one must also establish a Koszul self‐duality result for the right modules $$E_M$$, the configurations of disks in $$M$$.…”

“…The explicit cooperad structure on 𝐵(com) was utilized by Behrens to calculate the ring of 𝔽 2 Dyer-Lashof operations of the spectral Lie operad, which he used to give a novel calculation of the unstable homotopy groups of spheres up to the 19-stem [9]. † It is natural to ask if there is a diagram of operads in spectra that lifts the previous diagram of algebraic operads After much speculation, this was recently realized by Ching-Salvatore [17]. To complete the program of translating Poincaré-Koszul duality into a more classical form, one must also establish a Koszul self-duality result for the right modules 𝐸 𝑀 , the configurations of disks in 𝑀.…”

Unstable Periodic Homotopy Theory

“…Some other well-known models for the E n -operad are the little n-cubes operad [HM22, Example 1.8] and the Fulton-MacPherson operad [HM22, § §1.8-1.9]. We refer the reader to [BM23b] for an exposition about the E n -operad and to [CS22] for an application of various models for the E n -operad. The set…”