The bar construction as a Hopf algebra

Recall that deformations of Hochschild cochains are controlled by a homotopy Baues algebra (B∞-algebra) structure, and therefore a homotopy Gerstenhaber algebra structure on the Hochschild complex. The purpose of the present article is twofold. We first answer one of the classical questions addressed to algebraic topology. Namely, we construct a B∞-algebra structure on singular cochain complexes, that is a differential Hopf algebra structure on their bar construction. The question goes back to the work of Adams on the cobar construction and has been solved partially by Baues, who has constructed such a structure in the particular case of 1-reduced spaces. Our second purpose is to show that these structures are compatible with the Gerstenhaber and Schack cohomology comparison theorem. A strong version of this theorem asserts that there is a cochain equivalence between the Hochschild complex of a certain algebra and the usual singular cochain complex of a space. We prove that this cochain equivalence is induced by a morphism of B∞-algebras. On that ground, we point out that the meaning of the Deligne Hochschild cohomology conjecture can be understood differently as it is classically (e.g. in the work of Kontsevich-Soibelman). A.M.S Classification. 16E40; 55N10; 18D50; 55P48