On Mapping Spaces of Differential Graded Operads with the Commutative Operad as Target

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 MapO0 (En,C) with a cofibrant En-operad En on the source and the commutative operad C on the target. First, we prove that the homotopy class of a morphism φ : En → C is uniquely determined by a multiplicative constant which gives the action of φ on generating operations in homology. From this result, we deduce that the connected components of MapO0 (En,C) are in bijection with the ground ring. Then we prove that each of these connected components is contractible. In the case n = ∞, we deduce from our results that the space of homotopy selfequivalences of an E∞-operad in differential graded modules has contractible connected components indexed by the invertible elements of the ground ring.