In Cohen and Jones' paper on string topology(A homotopic realization of string topology), they explain the relation between loop homology and Hochschild cohomology of singular cochain complex.
They first state a theorem of loop cohomology and Hochschild homology, $$H^\ast(LM) \simeq HH_\ast(C^\ast(M))$$ But after that, they get chain complex by dualizing cochain complex. (which are not finite dimensional or finitely generated)
I googled on this and found Cohen and Voronov's notes on string topology, which says that the following are obvious identification $$Hom(C^\ast(X)^{\otimes q+1},k) \simeq Hom(C^\ast(X)^{\otimes q},C_\ast(X))$$ Is there any reason that chain complex of simply connected space is reflexive?
I first thought they may go around the minimal model, but they are using integer coefficient in the original paper (25p.).