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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Rational homotopy type of the classifying space for fibrewise self-equivalences
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by Urtzi Buijs and Samuel B. Smith PDF
Proc. Amer. Math. Soc. 141 (2013), 2153-2167 Request permission

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
  • Email: ubuijs@ub.edu
  • Samuel B. Smith
  • Affiliation: Department of Mathematics, Saint Joseph’s University, Philadelphia, Pennsylvania 19131
  • MR Author ID: 333158
  • Email: smith@sju.edu
  • 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
  • © Copyright 2012 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 2153-2167
  • MSC (2010): Primary 55P62, 55Q15
  • DOI: https://doi.org/10.1090/S0002-9939-2012-11560-6
  • MathSciNet review: 3034442