Formality theorem for Hochschild cochains via transfer
Abstract
We construct a 2-colored operad G+ which, on the one hand, extends the operad G governing homotopy Gerstenhaber algebras and, on the other hand, extends the 2-colored operad governing open-closed homotopy algebras (OCHA). We show that Tamarkin's G-structure on the Hochschild cochain complex C(A) of an A-infinity algebra A extends naturally to a G+ structure on the pair (C(A), A). We show that a formality quasi-isomorphism for the Hochschild cochains of the polynomial algebra can be obtained via transfer of this G+ structure to the cohomology of the pair (C(A), A). We show that G+ 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 G+ and E1(SC) are non-formal.