The Borel cohomology of the loop space of a homogeneous space

Let B ′ → f B ← p E 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 S 1 -equivariant cohomology of the space of the based loops on the complex projective space C P m , where the S 1 -action is induced by a linear action of S 1 on C P m .