On the A

It is proved that the associative differential graded algebra of (polynomial) polyvector fields on a vector space (may be infinite-dimensional) is quasi-isomorphic to the corresponding cohomological Hochschild complex of (polynomial) functions on this vector space as an associative differential graded algebra. This result is an A∞-version of the Formality conjecture of Maxim Kontsevich [K]. 1. Configuration spaces and their compactifications Let n,m be non-negative integers satisfying the inequality n + m ≥ 1. We define a configuration space Confn,m which will play the crucial role in the construction of the A∞-quasi-isomorphism F : T • poly → D • poly in Section 2. We set Confn,m = {(p1, . . . , pn; q1, . . . , qm) | pi ∈ R<0, qj ∈ R>0 and p1 < . . . < pn, q1 > . . . > qm}. Confn,m is a smooth manifold of dimension n+m. The group G (1) = {t 7→ at, a > 0, a ∈ R} acts on the space Confn,m. The quotient space Cn,m = Confn,m/G (1) is a manifold of dimension n+m−1. We are going to describe a compactification Cn,m of the space Cn,m, which is a manifold with corners in the sense of Maxim Kontsevich [K], Sect. 5.2. We defined also a manifold Cn, n ≥ 2: Confn = {(p1, . . . , pn) | pi ∈ R, p1 < . . . < pn}, G(2) = {t 7→ at+ b, a, b ∈ R, a > 0} and Cn = Confn/G (2), dimCn = n− 2. The construction of the compactification Cn is analogous to [K], Sect. 5.2. The strata CT are labeled by trees T and Cn = ⋃