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

Abstract

Let Aut($p$) denote the space of all self-fibre-homotopy equivalences of a fibration $p : E \to B$. 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: $$\pi_*(\rm{Aut} (p)) \otimes \mathbb{Q} \cong H_*(\rm{Der}_{\wedge V}(\wedge V \otimes \wedge W)).$$

Here $\wedge V \to \wedge V \otimes \wedge W$ is the Koszul-Sullivan model of the fibration and $\rm{Der}_{\wedge V} (\wedge{V} \otimes \wedge W)$ is the DG Lie algebra of derivations vanishing on $\wedge V$. We obtain related identifications of the rationalized homotopy groups of fibrewise mapping spaces and of the rationalization of the nilpotent group $\pi_0 (\rm{Aut}_\sharp(p)$ is a fibrewise adaptation of the submonoid of maps inducing the identity on homotopy groups.