Module derivations and non triviality of an evaluation fibration

We give a sufficient condition for the evaluation fibration, whose total space is the free iterated loop space, not to be totally non cohomologous to zero with respect to a given field. As an consequence, in the rational case, we generalize results of Vigue-Poirrier [3] and of Smith [2] on the cohomological triviality of the free loop fibration. We also deduce (in)equalities concerning the dimension of the mod $p$ reduction of the image by the Whitehead products on the $n$-th homotopy group of an $(n-1)$-connected space from some homological property of the space. The result recovers the inequality due to Chen [1].

[1] K. T. Chen, On the Whitehead products, Proc. Amer. Math. Soc. 34(1972), 257-259

[2] L. Smith, On the characteristic zero cohomology of the free loop space, Amer. J. Math. 103(1981), 887-910

[3] M. Vigue-Poirrier, Dan le fibre de l'espace des lacets libres, la fibre n'est pas, en general, totalement non cohomologue a zero, Math. Z. 181(1982), 537-542