A spectral sequence for spaces of maps between operads (Q6593008)

Let \(P,Q\) be topological or simplicial operads. This article build tools to compute the derived mapping space \(\mathrm{RHom}(P,Q)\), which can be seen as the space of operad morphisms \(P \to Q\) ``up to homotopy.'' The computation of these spaces is crucial in e.g., embedding calculus. To attack this problem, operads are modeled by dendroidal spaces (introduced by \textit{I. Moerdijk} and \textit{I. Weiss} [Algebr. Geom. Topol. 7, 1441--1470 (2007; Zbl 1133.55004)]) via the dendroidal nerve and thus the authors study derived mapping spaces \(\mathrm{RHom}(X,Y)\) between dendroidal spaces.\N\NA dendroidal space \(X\) can be truncated to operations of arity \(\leq k\) to obtain a truncated dendroidal space \(U_k X\). Such restrictions produce a tower of derived mapping spaces: \N\[\N\mathrm{RHom}(X,Y) \to \dots \to \mathrm{RHom}(U_k X, U_k Y) \to \mathrm{RHom}(U_{k-1} X, U_{k-1} Y) \to \dots \to \mathrm{RHom}(U_0 X, U_0 Y).\N\]\NSuch a construction is reminiscent of common constructions in embedding calculus. While spectral sequences do not appear, per se, outside of the title of this article, the reviewer presumes that the title refers to the spectral sequence that could be built out of this tower.\N\NThe first result of this article is that this tower ``converges,'' i.e., \(\mathrm{RHom}(X,Y)\) is weakly equivalent to the homotopy limit of the tower. The main result of the article is that \(\mathrm{RHom}(U_k X, U_k Y)\) can be expressed in terms of \(\mathrm{RHom}(U_{k-1} X, U_{k-1} Y)\) and a kind of latching-matching construction of \(X\) and \(Y\), for 1-reduced Segal dendroidal spaces.\N\NThis main result allows the study of the homotopy fiber of the restriction map \(\mathrm{RHom}(U_k X, U_k Y) \to \mathrm{RHom}(U_{k-1} X, U_{k-1} Y)\), which is thus equivalent to the homotopy fiber of the map between latching/matching constructions. As an application, for the little disks operads \(E_n\) and \(E_{n+d}\) (which are crucial in embedding calculus), the authors prove that the homotopy fiber of \(\mathrm{RHom}(U_k E_n, U_k E_{n+d}) \to \mathrm{RHom}(U_{k-1} E_n, U_{k-1} E_{n+d})\) is \(((k-1)(d-2)+1)\)-connected. Therefore, when \(d \geq 3\), the connectivity goes to infinity as \(k\) increases, which implies that the homotopy groups of \(\mathrm{RHom}(E_n, E_{n+d})\) -- which are connected to spaces of long knots -- can be computed from a finite stage of the tower. This is, again, reminiscent of other results in embedding calculus (see e.g., [\textit{B. Fresse} et al., ``The rational homotopy of mapping spaces of E${}_n$ operads'', Preprint, \url{arXiv:1703.06123}]).