Open Access
2010 The rational homotopy type of the space of self-equivalences of a fibration
Yves Félix, Gregory Lupton, Samuel B. Smith
Homology Homotopy Appl. 12(2): 371-400 (2010).


Let Aut($p$) denote the space of all self-fibre-homotopy equivalences of a fibration $p : E \to B$. 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: $$\pi_*(\rm{Aut} (p)) \otimes \mathbb{Q} \cong H_*(\rm{Der}_{\wedge V}(\wedge V \otimes \wedge W)).$$

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


Download Citation

Yves Félix. Gregory Lupton. Samuel B. Smith. "The rational homotopy type of the space of self-equivalences of a fibration." Homology Homotopy Appl. 12 (2) 371 - 400, 2010.


Published: 2010
First available in Project Euclid: 28 January 2011

zbMATH: 1214.55011
MathSciNet: MR2771595

Primary: 55P62 , 55Q15

Keywords: derivation‎ , Fibre-homotopy equivalence , function space , Samelson Lie algebra , Sullivan minimal model

Rights: Copyright © 2010 International Press of Boston

Vol.12 • No. 2 • 2010
Back to Top