arXiv: Algebraic Topology | 2019
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 $p_\\infty \\colon UE \\to \\mathrm{Baut}_1(X)$ 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.