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**1 - 1**of**1**### THE RATIONAL HOMOTOPY TYPE OF THE SPACE OF SELF-EQUIVALENCES OF A FIBRATION

, 2009

"... 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 ..."

Abstract
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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.