The cup product on Hochschild cohomology via twisting cochains and applications to Koszul rings

Given an acyclic twisting cochain π:C→A, from a curved dg coalgebra C to a dg algebra A, we show that the associated twisted hom complex Homkπ(C,A) has cohomology equal to the Hochschild cohomology of A, as a graded ring. As a corollary we find that the Hochschild cohomology of a Koszul algebra A, along with its cup product, is a subquotient of the tensor product algebra A!⊗A of A with its Koszul dual.