Formality Theorem for Hochschild Cochains via Transfer

Abstract

We construct a 2-colored operad Ger∞+ which, on the one hand, extends the operad Ger∞ governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy algebras. We show that Tamarkin's Ger∞-structure on the Hochschild cochain complex C•(A, A) of an A∞-algebra A extends naturally to a Ger∞+-structure on the pair (C•(A, A), A). We show that a formality quasi-isomorphism for the Hochschild cochains of the polynomial algebra can be obtained via transfer of this Ger∞+-structure to the cohomology of the pair (C•(A, A), A). We show that Ger∞+ is a sub DG operad of the first sheet E1(SC) of the homology spectral sequence for the Fulton-MacPherson version SC of Voronov's Swiss Cheese operad. Finally, we prove that the DG operads Ger∞+ and E1(SC) are non-formal. © 2011 Springer.