The rational homotopy type of the space of self-equivalences of a fibration

TL;DR

This paper explores the rational homotopy type of the space of self-equivalences of a fibration, providing algebraic models and identifications in rational homotopy theory for simply connected CW complexes.

Contribution

It identifies the rational Samelson Lie algebra of Aut(p) using DG Lie algebra derivations from the Koszul-Sullivan model, advancing understanding of automorphisms in rational homotopy theory.

Findings

Identified the rational Samelson Lie algebra of Aut(p).

Provided algebraic models for rational homotopy groups of fibrewise mapping spaces.

Described the rationalization of a nilpotent subgroup of Aut(p).

Abstract

Let Aut(p) denote the topological monoid of self-fibre-homotopy equivalences of a fibration p:E\to B. We make a general study of this monoid, especially in rational homotopy theory. When E and B are simply connected CW complexes with E finite, we identify the rational Samelson Lie algebra of the identity component of Aut(p) as the homology of a certain DG Lie algebra of derivations arising from the Koszul-Sullivan model of p. We obtain related identifications for the rational homotopy groups of fibrewise mapping spaces and for the rationalization of a natural nilpotent subgroup of Aut(p).