The rational homotopy type of the space of self-equivalences of a fibration

摘要

Let $mathrm{Aut}(p)$ denote the space of all self-fibre-homotopy equivalences of a fibration $p colon E o B$. When $E$ and $B$ are simply connected CW complexes with $E$ finite, we identify the rational Samelson Lie algebra of this monoid by means of an isomorphism: [ pi_*(mathrm{Aut}(p)) otimes mathbb{Q} cong H_*(mathrm{Der}_{land V}(land Votimes land W)). ] Here $land Vo land V otimes land W$ is the Koszul-Sullivan model of the fibration and $mathrm{Der}_{land V}(land Votimes land W)$ is the DG Lie algebra of derivations vanishing on $land V$. We obtain related identifications of the rationalized homotopy groups of fibrewise mapping spaces and of the rationalization of the nilpotent group $pi_0(mathrm{Aut}_sharp(p))$, where $mathrm{Aut}_sharp(p)$ is a fibrewise adaptation of the submonoid of maps inducing the identity on homotopy groups.