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

Let $\mathrm{Aut}(p)$ denote the space of all self-fibre-homotopy equivalences of a fibration $p \colon E \to B$. 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:

\[ \pi_*(\mathrm{Aut}(p)) \otimes \mathbb{Q} \cong H_*(\mathrm{Der}_{\land V}(\land V\otimes \land W)). \]

Here $\land V\to \land V \otimes \land W$ is the Koszul-Sullivan model of the fibration and $\mathrm{Der}_{\land V}(\land V\otimes \land W)$ is the DG Lie algebra of derivations vanishing on $\land V$. We obtain related identifications of the rationalized homotopy groups of fibrewise mapping spaces and of the rationalization of the nilpotent group $\pi_0(\mathrm{Aut}_\sharp(p))$, where $\mathrm{Aut}_\sharp(p)$ is a fibrewise adaptation of the submonoid of maps inducing the identity on homotopy groups.

Keywords

fibre-homotopy equivalence, Samelson Lie algebra, function space, Sullivan minimal model, derivation

2010 Mathematics Subject Classification

55P62, 55Q15

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Published 1 January 2010