Formality and rational homotopy theory of relative homotopy automorphisms

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iterated connected sums of complex projective 3-spaces.At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 3: Manuscript. Paper 4: Manuscript.