The cohomology of a pull-back on K-formal spaces

We prove a collapse theorem for the Eilenberg–Moore spectral sequence with coefficients in a field K converging to the cohomology of the pull-back of a fibration q :E→B by a continuous map f :X→B when E, X and B are K-formal. We also show that the cohomology algebra of the pull-back can be expressed via the torsion functor with the shc-minimal model for B in the sense of Ndombol and Thomas [Topology 41 (2002) 85] and its free extensions for E and X without the assumption of K-formality. Moreover not only does the shc-minimal models for E, B and X enable us to construct a model for the Eilenberg–Moore spectral sequence also they help in computing the spectral sequence.