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 → 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 isomor-phism: π * (Aut(p)) ⊗ Q ∼ = H * (Der ∧V (∧V ⊗ ∧W)). Here ∧V → ∧V ⊗ ∧W is the Koszul-Sullivan model of the fibra-tion and Der ∧V (∧V ⊗ ∧W) is the DG Lie algebra of derivations vanishing on ∧V. We obtain related identifications of the rationalized homotopy groups of fibrewise mapping spaces and of the rationalization of the nilpotent group π 0 (Aut (p)), where Aut (p) is a fibrewise adaptation of the submonoid of maps inducing the identity on homotopy groups.