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

Yves Félix, Gregory Lupton and Samuel B. Smith

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.

Homology, Homotopy and Applications, Vol. 12 (2010), No. 2, pp.371-400.

Available as: dvi dvi.gz ps ps.gz pdf