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 isomorphism: π∗(Aut(p)) ⊗ Q ∼= H∗(Der∧V (∧V ⊗ ∧W )). Here ∧V → ∧V ⊗ ∧W is the Koszul-Sullivan model of the fibration 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 π0(Aut♯(p)) where Aut♯(p) is a fibrewise adaptation of the submonoid of maps inducing the identity on homotopy groups.