# Bashar Saleh: Formality and rational homotopy theory of relative homotopy automorphisms

**Time: **
Fri 2020-10-23 13.00

**Location: **
Zoom, meeting ID: 611 6862 3282

**Doctoral student: **
Bashar Saleh

**Opponent: **
Aniceto Murillo, University of Malaga

**Supervisor: **
Alexander Berglund

**DiVA Link:
Click here!
**

### Abstract

This PhD thesis consists of four papers treating topics in rational homotopy theory.

In Paper I, we establish two formality conditions in characteristic zero. We prove that a dg Lie algebra is formal if and only if its universal enveloping algebra is formal. We also prove that a commutative dg associative algebra is formal as a dg associative algebra if and only if it is formal as a commutative dg associative algebra. We present some consequences of these theorems in rational homotopy theory.

In Paper II, which is coauthored with Alexander Berglund, we construct a dg Lie algebra model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a subspace, so called relative homotopy automorphisms.

In Paper III, which is coautohored with Hadrien Espic, we prove that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented and that the rationalization map from this group to its rational analogue has a finite kernel.

In Paper IV, we study rational homological stability for the classifying space of the monoid of homotopy automorphisms of iterated connected sums of complex projective 3-spaces.