Formality Theorem for Hochschild Cochains via Transfer

ISSN： 0377-9017

Source： Letters in Mathematical Physics, Vol.97, Iss.2, 2011-08, pp. : 109-149

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

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 $${{bf Ger}^+_{infty}}$$ -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 $${{bf Ger}^+_{infty}}$$ -structure to the cohomology of the pair (C ^•(A, A), A). We show that $${{bf Ger}^+_{infty}}$$ is a sub DG operad of the first sheet E ¹(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 $${{bf Ger}^+_{infty}}$$ and E ¹(SC) are non-formal.