On mapping spaces of differential graded operads with the commutative operad as target (Q2428788)

The category of dg-operads has a model structure and so there is a natural notion of simplicial mapping space between dg-operads. These mapping spaces are studied in this paper for the case where the source is a non-unitary version of a cofibrant \(E_n\)-operad and the target is the commutative operad. The main result of the paper is that these mapping spaces have contractible connected components indexed by the elements of the ground ring. To show this, the author uses the model for cofibrant \(E_n\)-operads developed in [\textit{B. Fresse}, ``Koszul duality of \(E_n\)-operads'', Sel. Math., New Ser. 17, No. 2, 363--434 (2011; Zbl 1248.55003)], given by an operadic cobar construction of a desuspension of the dual cooperad of an \(E_n\)-operad. This has a filtration by arity of generators and this is used to decompose the mapping spaces into towers of fibrations with Eilenberg-MacLane spaces as fibers. The Bousfield-Kan extended homotopy spectral sequence is applied and the main result follows from the fact that this almost vanishes at the \(E_1\) stage. As a corollary, it is shown that the space of homotopy automorphisms of a cofibrant \(E_\infty\)-operad in dg-modules has contractible connected components indexed by the units in the ground ring. The author comments that the corresponding homotopy automorphisms seem to be more complicated for \(n<\infty\) and notes an expected relation to the Grothendieck-Teichmüller group in the case \(n=2\).