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Rational homotopy theory of mapping spaces via Lie theory for Linfinity algebras
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THE RATIONAL HOMOTOPY TYPE OF THE SPACE OF SELFEQUIVALENCES OF A FIBRATION
, 2009
"... Let Aut(p) denote the space of all selffibrehomotopy 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 selffibrehomotopy 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 KoszulSullivan 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.