Compatibility with Cap-Products in Tsygan’s Formality and Homological Duflo Isomorphism

Abstract

Abstract. In this paper we prove, with details and in full generality, that the isomorphism induced on tangent homology by the Shoikhet-Tsygan formality L∞-quasi-isomorphism for Hochschild chains is compatible with capproducts. This is a homological analog of the compatibility with cup-products of the isomorphism induced on tangent cohomology by Kontsevich formality L∞-quasi-isomorphism for Hochschild cochains.As in the cohomological situation our proof relies on a homotopy argument involving a variant of Kontsevich eye. In particular we clarify the rôle played by the I-cube introduced in [4].Since we treat here the case of a most possibly general Maurer-Cartan element, not forced to be a bidifferential operator, then we take this opportunity to recall the natural algebraic structures on the pair of Hochschild cochain and chain complexes of an A∞-algebra. In particular we prove that they naturally inherit the structure of an A∞-algebra with an A∞-(bi)module.

Căldăraru's conjecture and Tsygan's formality

“…While these are a priori only L ∞ -quasi-isomorphisms they are nonetheless compatible with products up to homotopy [16], [7] in a strong explicit sense, and this turns out to be enough for our purposes.…”

“…The proofs of Theorems 1.1 and 1.3 depend on the simultaneous globalization of a number of local formality results due to Kontsevich [16] (see also [17]), Tsygan [25], Shoikhet [20] and the first two authors [8], [6], [7]. This globalization is performed by a functorial version of formal geometry [4] (see also [28]).…”

“…L-differential operators over R. In this section we define the algebra of poly-differential operators of a Lie algebroid and we list some of its properties. We give some explicit formulae along the lines of [7].…”

“…We will do this without relying on explicit formulae (as they are quite complicated). Instead we will reduce to a relative version of [7] that discusses Hochschild (co)homology. Explicit formulae are given in Appendix B.…”

“…A complete treatment in a very general setting was given in [7]. It is shown that the precalculus structure can be obtained from two interacting B ∞ -module structures on C • (A).…”