The Borel cohomology of the loop space of a homogeneous space

Let B′→fB←pE be a diagram in which p is a fibration and the pair (f,p) of the maps is relatively formalizable. Then, we show that the rational cohomology algebra of the pullback of the diagram is isomorphic to the torsion product of algebras H⁎(B′) and H⁎(E) over H⁎(B). Let M be a space which admits an action of a Lie group G. The isomorphism of algebras enables us to represent the cohomology of the Borel construction of the space of free (resp. based) loops on M in terms of the torsion product if M is equivariantly formal (resp. G-formal). Moreover, we compute explicitly the S1-equivariant cohomology of the space of the based loops on the complex projective space CPm, where the S1-action is induced by a linear action of S1 on CPm.