We calculate the higher homotopy groups of the Deligne-Getzler ∞-groupoid associated to a nilpotent L∞-algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.

8.1. κ-classes and the stable cohomology of the diffeomorphism group 155 8.2. Borel's calculation of the stable cohomology of arithmetic groups 156 8.3. Relation between Borel classes and κ-classes . .

Abstract. We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. The main difficulty is to find a suitable notion of algebra homotopy that generalizes to algebras over operads O. To solve this problem, we introduce thick maps of O-algebras and special thick maps that we call pseudo-derivations that serve as appropriate generalizations of algebra homotopies for the purposes of homological perturbation theory.As an application, we derive explicit formulas for transferring Ω(C)-algebra structures along contractions, where C is any connected cooperad in chain complexes. This specializes to transfer formulas for O∞-algebras for any Koszul operad O, in particular for A∞, C∞, L∞ and G∞-algebras. A key feature is that our formulas are expressed in terms of the compact description of Ω(C)-algebras as coderivation differentials on cofree C-coalgebras. Moreover, we get formulas not only for the transferred structure and a structure on the inclusion, but also for structures on the projection and the homotopy.

We prove that a nilpotent space is both formal and coformal if and only if it is rationally homotopy equivalent to the derived spatial realization of a graded commutative Koszul algebra. We call such spaces Koszul spaces and we show that the rational homotopy groups and the rational homology of iterated loop spaces of Koszul spaces can be computed by applying certain Koszul duality constructions to the cohomology algebra.