The cohomology of a pull-back on ${\mathbb{K}}$-formal spaces

We prove a collapse theorem for the Eilenberg-Moore spectral sequence with coefficients in a field ${\mathbb{K}}$ converging to the cohomology of the pull-back of a fibration $q :E \to B$ by a continuous map $f : X \to B$ when $E$, $X$ and $B$ are ${\mathbb{K}}$-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 [N-T] and its free extensions for $E$ and $X$ without the assumption of ${\mathbb{K}}$-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.

[N-T] B. Ndombol and J. -C. Thomas, On the cohomology algebra of free loop spaces, to appear in Topology.