Rational Homotopy Type of the Classifying Space for Fibrewise Self-Equivalences

TL;DR

This paper develops a differential graded Lie algebra model for the classifying space of fibrewise self-equivalences of a fibration, enabling classification of their rational homotopy types and conditions for double loop-space structures.

Contribution

It introduces a novel differential graded Lie algebra model for Baut_1(p) and applies it to classify rational homotopy types and analyze loop-space properties.

Findings

Provides a differential graded Lie algebra model for Baut_1(p)

Classifies rational homotopy types of Baut_1(p)

Identifies conditions for aut_1(p) to be a double loop-space after rationalization

Abstract

Let p be a fibration of simply connected CW complexes with finite base B and fibre F. Let aut_1(p) denote the identity component of the space of all fibre-homotopy self-equivalences of p and Baut_1(p) the classifying space for this topological monoid. We give a differential graded Lie algebra model for Baut_1(p). We use this model to give classification results for the rational homotopy types represented by Baut_1(p) and also to obtain conditions under which the monoid aut_1(p) is a double loop-space after rationalization.