Rational homotopy type of the classifying space for fibrewise self-equivalences

Buijs, Urtzi; Smith, Samuel

doi:10.1090/S0002-9939-2012-11560-6

Rational homotopy type of the classifying space for fibrewise self-equivalences
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by Urtzi Buijs and Samuel B. Smith

Proc. Amer. Math. Soc. 141 (2013), 2153-2167

DOI: https://doi.org/10.1090/S0002-9939-2012-11560-6

Published electronically: December 13, 2012

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Abstract:

Let $p \colon E \to B$ be a fibration of simply connected CW complexes with finite base $B$ and fibre $F$. Let ${\mathrm {aut}}_1(p)$ denote the identity component of the space of all fibre-homotopy self-equivalences of $p$. Let ${\mathrm {Baut}}_1(p)$ denote the classifying space for this topological monoid. We give a differential graded Lie algebra model for ${\mathrm {Baut}}_1(p)$, connecting the results of recent work by the authors and others. We use this model to give classification results for the rational homotopy types represented by ${\mathrm {Baut}}_1(p)$ and also to obtain conditions under which the monoid ${\mathrm {aut}}_1(p)$ is a double loop-space after rationalization.

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