Koszul self-duality of manifolds (Q6564513)

The essence of Koszul duality is encapsulated in the commutativity of the diagram of algebraic operads.\N\[\N\begin{array} [c]{ccccc} s_{n}\mathrm{lie} & \rightarrow & \mathrm{pois}_{n} & \rightarrow & \mathrm{com}\\\N\mid\simeq & & \mid\simeq & & \mid\simeq\\\Ns_{n}K\left( \mathrm{com}\right) & \rightarrow & s_{n}K\left( \mathrm{pois}_{n}\right) & \rightarrow & K\left( \mathrm{lie}\right) \end{array}\N\]\NSince \(H_{\ast}\left( E_{n}\right) \cong\mathrm{pois}_{n}\) [\textit{F. R. Cohen} et al., The homology of iterated loop spaces. Springer, Cham (1976; Zbl 0334.55009)], it is seen that the bar-cobar duality of \(E_{n}\)-algebras is reflected as a theorem about Koszul self-duality of the homology of \(E_{n}\). Koszul duality was originally formulated for operads in \(\left( \mathrm{DGVect}_{\mathbb{Q}},\otimes\right) \), but was generalized to operads in \(\left( \mathrm{Top}_{\ast},\wedge\right) \)\ and \(\left( \mathrm{Sp},\wedge\right) \)\ by Ching [\textit{M. Ching}, Geom. Topol. 9, 833--933 (2005; Zbl 1153.55006)] and Salvatore [\url{https://www.mathgenealogy.org/id.php?id=92632}] independently. \textit{G. Arone} and \textit{M. Ching} [Operads and chain rules for the calculus of functors. Paris: Société Mathématique de France (SMF) (2011; Zbl 1239.55004)] used Koszul duality to endow the Goodwille derivatives \(\partial_{\ast}F\)\ of a functor \(F:\mathrm{Top} \rightarrow\mathrm{Top}\) with the structure of a \(\mathrm{lie}:=K\left( \mathrm{com}\right) \)\ bimodule.\N\NIt is natural to ask whether there is a diagram of operads in spectra that lifts the previous diagram of algebraic operads\N\[\N\begin{array} [c]{ccccc} s_{n}\mathrm{lie} & \rightarrow & \Sigma_{+}^{\infty}E_{n} & \rightarrow & \mathrm{com}\\\N\mid\simeq & & \mid\simeq & & \mid\simeq\\\Ns_{n}K\left( \mathrm{com}\right) & \rightarrow & s_{n}K\left( \Sigma _{+}^{\infty}E_{n}\right) & \rightarrow & K\left( \mathrm{lie}\right) \end{array}\N\]\Nwhich was realized by \textit{M. Ching} and \textit{P. Salvatore} [Proc. Lond. Math. Soc. (3) 125, No. 1, 1--60 (2022; Zbl 1529.55012)] after much speculation. 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 \(M\). This paper constructs a compatible diagram of right modules that witnesses the compactly supported Koszul self-duality of \(E_{M}\).\N\NChing [\url{https://mching.people.amherst.edu/Work/skye.pdf}] noticed the stabilized configuration spaces of framed manifolds could be endowed with the structure of a shifted Lie right module in two ways, namely, one through Goodwillie calculus of the functor \(X\rightarrow\Sigma^{\infty}\mathrm{Map}\left( M^{+},X\right) \)\ and one through manifold calculus combined with the hypothesized self-duality of \(E_{n}\), conjectureing that these two right module structures should coincide. The author proves this as a consequence of the self-duality of \(E_{M}\), the main work for which is to find a way to access the operad equivalence\N\[\N\Sigma_{+}^{\infty}E_{n}\simeq s_{n}K\left( \Sigma_{+}^{\infty}E_{n}\right)\N\]\NThis paper interprets Koszul duality in terms of Spivak normal fibrations instead. The Koszul dualizing fibration of an operad is defined, being of the structure of an operad in parametrized spectra. Using Verdier duality, it is shown that the Thom complex of the dualizing fibration is a model of the Koszul dual of \(O\). The author then shows that self-duality is equivalent to trivializing the Koszul dualizing fibration as an operad in parametrized spectra. This procedure lifts the homological theory of Poincaré-Koszul operads of \textit{C. Malin} [Homology Homotopy Appl. 26, No. 1, 229--258 (2024; Zbl 1546.55002)] to operads in spectra, analoguous to how Atiyah duality lifts Poincaré duality in (co)homology to a statement in spectra.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] introduces (co)operads and right (co)modules using partial (de)composites, recalling two models of the \(E_{n}\)\ operad and the right modules \(E_{M}\), as well as an extension of this construction to the one-point compactification \(M^{+}\).\N\N\item[\S 3] recounts a conjecture of Ching concerning manifold calculus and Goodwille calculus.\N\N\item[\S 4] gives a review of parametrized spectra.\N\N\item[\S 5] establishes Verdier duality for the functor of relative sections \(\Gamma^{A}\left( -\right) \).\N\N\item[\S 6] reviews the Koszul duality of \textit{M. Ching} and \textit{P. Salvatore} [Proc. Lond. Math. Soc. (3) 125, No. 1, 1--60 (2022; Zbl 1529.55012)], constructing the Koszul dualizing fibration \(\xi_{O}\)\ associated to an operad in unpointed spaces. Concerning an Koszul duality in terms of Thom complexes \(\mathrm{Th}\left( -\right) \)\ of operads and right modules in the category of parametrized spectra, we have (Propositions 6.14 and 6.32)\N\NProposition. To an operad \(O\)\ and right module pair \(\left( R,A\right) \)\ in \(\left( \mathrm{Top},\times\right) \), we can associate an operad \(\xi_{O}\)\ and right module \(\xi_{\left( R,A\right) }\)\ in \(\left( \mathrm{ParSp},\overline {\wedge}\right) \)\ for which there are compatible equivalences\N\begin{align*}\N& \mathrm{Th}\left( \xi_{O}\right) \overset{\simeq}{\rightarrow}K\left( \Sigma_{+}^{\infty}O\right) \\\N& \mathrm{Th}\left( \xi_{\left( R,A\right) }\right) \overset{\simeq }{\rightarrow}K\left( \Sigma^{\infty}R/A\right)\N\end{align*}\N\N\item[\S 7] proves that codimension 0 submanifolds of \(\mathbb{R}^{n} \)\ satisfy noncompact Koszul self-duality.\N\N\item[\S 8] addresses Weiss cosheaves taking value in the category of right modules over an operad, which enables the author to extend the proposition of \S 6 to all tame, framed manifolds.\N\N\item[\S 9] establishes the main result that \(E_{M}\)\ is Koszul self-dual, giving applications. The following three theorems are established.\N\NTheorem 9.1 (Koszul self-duality of \(E_{M}\)). For a framed \(n\)-manifold \(M\), there are equivalences compatible with the self-duality of \(E_{n}\)\N\[\N\Sigma_{+}^{\infty}E_{M}\simeq S_{\left( n,n\right) }K\left( \Sigma ^{\infty}E_{M^{+}}\right)\N\]\N\NTheorem 9.3 (Pontryagin-Thom equivalence). For framed \(n\)-manifolds \(M,N\), there is a map\N\[\N\mathrm{Map}_{\Sigma_{+}^{\infty}E_{n}}^{h}\left( \Sigma_{+}^{\infty} E_{M},\Sigma_{+}^{\infty}E_{N}\right) \rightarrow\mathrm{Map}_{\Sigma _{+}^{\infty}E_{n}}^{h}\left( \Sigma^{\infty}E_{N^{+}},\Sigma^{\infty }E_{M^{+}}\right)\N\]\N\NTheorem 9.5 (Poincaré-Koszul duality for left \(\Sigma_{+}^{\infty}E_{n} \)-modules). For a framed \(n\)-manifold \(M\) and a left \(\Sigma_{+}^{\infty}E_{n}\)-module \(L\), there is an equivalence\N\[\N\int\nolimits_{\Sigma_{+}^{\infty}E_{M}}L\overset{\simeq}{\rightarrow} \int\nolimits^{S_{\left( n,n\right) }\Sigma^{\infty}E_{M^{+}}^{\vee}}B\left( L\right)\N\]\N\N\end{itemize}