  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_*(\Omega X) is isomorphic to
the free loop space cohomology H^{*}(X^{S^{1}}). We proved that this
isomorphism is compatible with both the cup product on HH^{*}S_*(\Omega X) and
on H^{*}(X^{S^{1}}). In particular, we explicit the algebra H^{*}(X^{S^{1}})
when X is a suspended space, a complex projective space or a finite CW-complex
of dimension p such that \frac {1}{(p-1)!}\in k.
