Mathematical Research Letters

Volume 18 (2011)

Number 2

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

Pages: 215 – 230

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n2.a3

Author

Benoit Fresse (UMR 8524 du CNRS et de l'Université Lille 1 - Sciences et Technologies, Cité Scientifique – Bâtiment M2, F-59655 Villeneuve d'Ascq Cédex (France))

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 $\Map_{\Op_0}(\EOp_n,\COp)$ with a cofibrant $E_n$-operad $\EOp_n$ on the source and the commutative operad $\COp$ on the target. First, we prove that the homotopy class of a morphism $\phi: \EOp_n\rightarrow\COp$ 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 $\Map_{\Op_0}(\EOp_n,\COp)$ are in bijection with the ground ring. Then we prove that each of these connected components is contractible. In the case $n = \infty$, we deduce from our results that the space of homotopy self-equivalences of an $E_\infty$-operad in differential graded modules has contractible connected components indexed by the invertible elements of the ground ring.

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