We consider in parallel pointed homotopy automorphisms of iterated wedge sums of finite CW–complexes and boundary-relative homotopy automorphisms of iterated connected sums of manifolds minus a disk. Under certain conditions on the spaces and manifolds, we prove that the rational homotopy groups of these homotopy automorphisms form finitely generated FI–modules, and thus satisfy representation stability for symmetric groups in the sense of Church and Farb. We also calculate explicit bounds on the weights and stability degrees of these FI–modules.

We obtain restrictions on the rational homotopy types of mapping spaces and of classifying spaces of homotopy automorphisms by means of the theory of positive weight decompositions. The theory applies, in particular, to connected components of holomorphic maps between compact Kähler manifolds as well as homotopy automorphisms of Kähler manifolds.

We construct a dg" role="presentation" style="display: inline; line-height: normal; font-size: 17.3333px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(64, 64, 64); font-family: "Times New Roman", Times, serif; position: relative;">dgdg Lie algebra model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a given subspace. We derive the model from a known model for based homotopy automorphisms together with general result on rational models for geometric bar constructions.

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.

This licentiate thesis consists of two papers treating subjects in rational homotopy theory.

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

In Paper II, we construct a differential graded Lie model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a subspace.

Over a field of characteristic zero we prove two formality conditions. 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 algebra is formal as a dg associative algebra if and only if it is formal as a commutative dg algebra. We present some consequences of these theorems in rational homotopy theory.

We study rational homological stability for the classifying space of the monoid of homotopy automorphisms of iterated connected sums of complex projective 3-spaces.

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.