The universal fibration with fibre X in rational homotopy theory

Abstract

Let X be a simply connected space with finite-dimensional rational homotopy groups. Let $$p_\infty :UE \rightarrow B\mathrm {aut}_1(X)$$ be the universal fibration of simply connected spaces with fibre X. We give a DG Lie algebra model for the evaluation map $$ \omega :\mathrm {aut}_1(B\mathrm {aut}_1(X_\mathbb {Q})) \rightarrow B\mathrm {aut}_1(X_\mathbb {Q})$$ expressed in terms of derivations of the relative Sullivan model of $$p_\infty $$. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space $$B\mathrm {aut}_1(X_\mathbb {Q})$$ as a consequence. We also prove that $$\mathbb {C} P^n_\mathbb {Q}$$ cannot be realized as $$B\mathrm {aut}_1(X_\mathbb {Q})$$ for $$n \le 4$$ and X with finite-dimensional rational homotopy groups.