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

Abstract

Let p : E → B p \colon E \to B be a fibration of simply connected CW complexes with finite base B B and fibre F F . Let a u t 1 ( p ) {\mathrm {aut}}_1(p) denote the identity component of the space of all fibre-homotopy self-equivalences of p p . Let B a u t 1 ( p ) {\mathrm {Baut}}_1(p) denote the classifying space for this topological monoid. We give a differential graded Lie algebra model for B a u t 1 ( p ) {\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 B a u t 1 ( p ) {\mathrm {Baut}}_1(p) and also to obtain conditions under which the monoid a u t 1 ( p ) {\mathrm {aut}}_1(p) is a double loop-space after rationalization.