The universal fibration with fibre $X$ in rational homotopy theory

TL;DR

This paper develops a DG Lie model for the universal fibration in rational homotopy theory, providing formulas for rational Gottlieb groups and showing certain projective spaces cannot be realized as classifying spaces.

Contribution

It introduces a DG Lie model for the evaluation map in rational homotopy theory and derives new formulas for rational Gottlieb groups and evaluation subgroups.

Findings

DG Lie model for evaluation map expressed via derivations

Formulas for rational Gottlieb groups and evaluation subgroups

Proof that certain rational projective spaces cannot be classifying spaces

Abstract

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