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Meeting ID: 611 6862 3282

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#### Title of the thesis:

Formality and rational homotopy theory of relative homotopy automorphisms

#### Author of the thesis:

Bashar Saleh

#### Opponent:

Aniceto Murillo (University of Malaga)

#### Supervisor:

Alexander Berglund

#### Abstract:

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.

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Meeting ID: 611 6862 3282

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