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 $p_{\infty }:UE\to {\mathrm{Baut}}_1(X)$ be the universal fibration of simply connected spaces with fibre $X$. We give a DG Lie model for the evaluation map $\omega :{\mathrm{aut}}_1({\mathrm{Baut}}_1(X_{\mathbb{Q}}))\to {\mathrm{Baut}}_1(X_{\mathbb{Q}})$ 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 ${\mathrm{Baut}}_1(X_{\mathbb{Q}})$ as a consequence. We also prove that $\mathbb{C}{P}_{\mathbb{Q}}^n$ cannot be realized as ${\mathrm{Baut}}_1(X_{\mathbb{Q}})$ for $n\le 4$ and $X$ with finite-dimensional rational homotopy groups.

15 pages