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Let FM be the universal fibration having fibre M an elliptic space with vanishing odd rational cohomology. We consider the rational cohomology of the total space of FM by using a function space model due to Haefliger, Brown and Szczarba when Halperin’s conjecture is affirmatively solved for the fibre. The calculation enables one to deduce that the cohomology of the classifying space of the self-homotopy equivalences of a c-symplectic space M is generated only by the Kedra-McDuff μ-classes if the cohomology of M is generated by a single element.

On the rational cohomology of the total space of the universal fibration with an elliptic fibre