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Koszul duality of E_n-operads

Research paper by Benoit Fresse

Indexed on: 14 Jan '10Published on: 14 Jan '10Published in: Mathematics - Algebraic Topology



Abstract

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 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 of operads 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-1 --> E_n at the level of cobar constructions.