Rational homotopy automorphisms of E_2-operads and the Grothendieck-Teichmüller group

by Benoit Fresse
Result announcement.


Abstract:

The singular chain complex of the little 2-cubes operad defines an operad in simplicial cocommutative coalgebras. This operad has a dual structure, formed by a cooperad in cosimplicial commutative algebras, which defines model of the prounipotent completion of the topological operad of little 2-cubes. The group of homotopy automorphisms of an object in a model category is formed by the homotopy classes of self homotopy equivalences of a cofibrantfibrant replacement of this object. We prove by using the model of cooperads in cosimplicial commutative algebras that the group of homotopy automorphisms of the rational prounipotent completion of the little 2-cubes operad is the Grothendieck-Teichmüller group over Q. The notes available on this page give a detailed account of our result. The proof will be integrated in a monograph in preparation on homotopy automorphisms of operads.

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(7/3/2010)