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

Authors: Urtzi Buijs and Samuel B. Smith
Journal: Proc. Amer. Math. Soc. 141 (2013), 2153-2167
MSC (2010): Primary 55P62, 55Q15
Published electronically: December 13, 2012
MathSciNet review: 3034442
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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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Additional Information

Urtzi Buijs
Affiliation: Departament d’Àlgebra i Geometria, Universitat de Barcelona, Gran Via de les Corts Catalanes, 585, 08007 Barcelona, Spain

Samuel B. Smith
Affiliation: Department of Mathematics, Saint Joseph’s University, Philadelphia, Pennsylvania 19131
MR Author ID: 333158

Keywords: Fibre-homotopy equivalences, classifying space, Quillen model, function space, Lie derivation
Received by editor(s): July 26, 2011
Received by editor(s) in revised form: September 18, 2011
Published electronically: December 13, 2012
Additional Notes: The first author was partially supported by the Ministerio de Ciencia e Innovación grant MTM2010-15831 and by the Junta de Andalucía grant FQM-213.
Communicated by: Brooke Shipley
Article copyright: © Copyright 2012 American Mathematical Society