Hochschild cohomology is topological

Let E be an r-connected finite complex of dimension n, where r≥1, and let p be an odd prime number such that rp≥n. A theorem of Anick implies that the cochain algebra image is equivalent to a commutative cochain algebra image. We prove that the Hochschild cohomology algebra of image is isomorphic as an algebra to the mod-p cohomology algebra of the free loop space on E.