Koszul duality of En-operads

The goal of this paper is to prove a Koszul duality result for E (n) -operads in differential graded modules over a ring. The case of an E (1)-operad, which is equivalent to the associative operad, is classical. For n > 1, the homology of an E (n) -operad is identified with the n-Gerstenhaber operad and forms another well-known Koszul operad. Our main theorem asserts that an operadic cobar construction on the dual cooperad of an E (n) -operad E-n defines a cofibrant model of E-n. This cofibrant model gives a realization at the chain level of the minimal model of the n-Gerstenhaber operad arising from Koszul duality. Most models of E (n) -operads in differential graded modules come in nested sequences E-1 subset of E-2 subset of ......subset of E-infinity homotopically equivalent to the sequence of the chain operads of little cubes. In our main theorem, we also define a model of the operad embeddings En-1 -> E-n at the level of cobar constructions.