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

Abstract: The category of differential graded operads is a cofibrantly generated model category and as such inherits simplicial mapping spaces. The vertices of an operad mapping space are just operad morphisms. The 1-simplices represent homotopies between morphisms in the category of operads. The goal of this paper is to determine the homotopy of the operadic mapping spaces Map(E_n,C) with a cofibrant E_n-operad on the source and the commutative operad on the target. First, we prove that the homotopy class of a morphism phi: E_n -> C is uniquely determined by a multiplicative constant which gives the action of phi on generating operations in homology. From this result, we deduce that the connected components of Map(E_n,C) are in bijection with the ground ring. Then we prove that each of these connected components is contractible. In the case where n is infinite, we deduce from our results that the space of homotopy self-equivalences of an E-infinity-operad in differential graded modules has contractible connected components indexed by invertible elements of the ground ring.