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{mathtt{E}_n} defines a cofibrant model of E_n{mathtt{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 E₁ ì E₂ ì ? ì E_￥{mathtt{E}_1subsetmathtt{E}_2subsetcdotssubsetmathtt{E}_{infty}} 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 E_n-1hookrightarrowE_n{mathtt{E}_{n-1}hookrightarrowmathtt{E}_n} at the level of cobar constructions.