Bar Complexes and Formality of Pull-backs

In this note, we show that the pull-back of a fibration by a formal map is formal. The fibration is required to be totally non-homologous to zero, and to be a formal map as well. We are here referring to the notion of formality in the setting of rational homotopy theory. This result extends a theorem of Vigué-Poirrier, [10], where it is proved that the fibre of such a fibration is a formal space. Our proof makes use of bar complexes, which, when we use a normalization due to Chen, become commutative differential graded algebras useful for rational homotopy theory. We conclude with an example which generalizes a result of Baum and Smith, [2].