The cohomology ring of free loop spaces

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Let X be a simply connected space and k a commutative ring. Goodwillie, Burghelea and Fiedorowiscz proved that the Hochschild cohomology of the singular chains on the pointed loop space HH*S*(ΩX) is isomorphic to the free loop space cohomology H*(Xˢ¹). We proved that this isomorphism is compatible with both the cup product on HH*S*(ΩX) and on H*(Xˢ¹). In particular, we explicit the algebra H*(Xˢ¹) when X is a suspended space, a complex projective space or a finite CW-complex of dimension p such that 1/(p-1)!∈ k.