Un arrangement central A est un ensemble fini de sous-espaces vectoriels dans un espace vectoriel complexe V de dimension finie. L'espace topologique complémentaire M(A) est l'ensemble des points de V qui n'appartiennent à aucun des sous-espaces de A. Dans ce travail, nous étudions la topologie de l'espace M(A) du point de vue de l'homotopie rationnelle. L'outil clé qui a servi de départ à cette thèse est un modèle rationnel de M(A) qui s'avère relativement simple à manipuler. À l'aide de ce modèle, nous obtenons plusieurs résultats sur la topologie de M(A). Citons par exemple des formules de récursion qui permettent de calculer certains invariants topologiques, dont les nombres de Betti, une preuve du fait que la caractéristique d'Euler de l'espace M(A) est nulle ou encore une description des arrangements (vérifiant une condition technique) dont le complémentaire est un wedge rationnel de sphères. Enfin, les résultats principaux de cet ouvrage sont une caractérisation des arrangements dont le complémentaire a le type d'homotopie d'un produit de sphères, et la preuve du fait que si le complémentaire n'est pas un produit de sphères, alors son algèbre de Lie d'homotopie contient la sous-algèbre de Lie libre à deux générateurs.

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. / <p>At the time of the doctoral defense, the following paper was unpublished and had a status as follows: Paper 2: Manuscript.</p>

If X is a simply connected CW complex, then it has a unique (up to isomorphism) minimal Sullivan model. There is an important rational homotopy invariant, called the rational Lusternik–Schnirelmann of X, denoted cat0(X), which has an algebraic formulation in terms of the minimal Sullivan model of X. We study another such numerical invariant called the rational retraction index of X, denoted r0(X), which is defined in terms of the minimal Sullivan model of X and satisfies 0 ≤ r0(X) ≤ cat0(X). It was introduced by Cuvilliez et al. as a tool to estimate the rational Lusternik–Schnirelmann category of the total space of a fibration. In this thesis we compute the rational retraction index on a range of rationally elliptic spaces, including for example spheres, complex projective space, the biquotient Sp(1) \ Sp(3) / Sp(1) × Sp(1), the homogeneous space Sp(3)/U(3) and products of these. In particular, we focus on formal spaces and formulate a conjecture to answer a question posed in the original article of Cuvilliez et al., “If X is formal, what invariant of the algebra H∗(X;Q) is r0(X)?”

If X is a simply connected CW complex, then it has a unique (up to isomorphism) minimal Sullivan model. There is an important rational homotopy invariant, called the rational Lusternik–Schnirelmann of X, denoted cat0(X), which has an algebraic formulation in terms of the minimal Sullivan model of X. We study another such numerical invariant called the rational retraction index of X, denoted r0(X), which is defined in terms of the minimal Sullivan model of X and satisfies 0 ≤ r0(X) ≤ cat0(X). It was introduced by Cuvilliez et al. as a tool to estimate the rational Lusternik–Schnirelmann category of the total space of a fibration. In this thesis we compute the rational retraction index on a range of rationally elliptic spaces, including for example spheres, complex projective space, the biquotient Sp(1) \ Sp(3) / Sp(1) × Sp(1), the homogeneous space Sp(3)/U(3) and products of these. In particular, we focus on formal spaces and formulate a conjecture to answer a question posed in the original article of Cuvilliez et al., “If X is formal, what invariant of the algebra H∗(X;Q) is r0(X)?”

Two standard invariants used to study the fundamental group of the complement X of a hyperplane arrangement are the Malcev completion of its fundamental group G and the cohomology groups of X with coefficients in rank one local systems. In this thesis, we develop a tool that unifies these two approaches. This tool is the Malcev completion S_p of G relative to a homomorphism p from G into (C^*)^N. The relative completion S_p is a prosolvable group that generalizes the classical Malcev completion; when p is the trivial representation, S_p is the Malcev completion of G. The group S_p is tightly controlled by the cohomology groups H^1(X,L_{p^k}) with coefficients in the irreducible local systems L_{p^k} associated to the representation p.The pronilpotent Lie algebra u_p of the prounipotent radical U_p of S_p has been described by Hain. If p is the trivial representation, then u_p is the holonomy Lie algebra, which is well-known to be quadratically presented. In contrast, we show that when X is the complement of the braid arrangement in complex two-space, there are infinitely many representations p from G into (C^*)^2 for which u_p is not quadratically presented.We show that if Y is a subtorus of the character torus T containing the trivial character, then S_p is combinatorially determined for general p in Y. We do not know whether S_p is always combinatorially determined. If S_p is combinatorially determined for all characters p of G, then the characteristic varieties of the arrangement X are combinatorially determined.When Y is an irreducible subvariety of T^N, we examine the behavior of S_p as p varies in Y. We define an affine group scheme S_Y over Y such that if Y = {p}, then S_Y is the relative Malcev completion S_p. For each p in Y, there is a canonical homomorphism of affine group schemes from S_p into the affine group scheme which is the restriction of S_Y to p. This is often an isomorphism. For example, if there exists p in Y whose image is Zariski dense in G_m^N, then this homomorphism is an isomorphism for general p in Y. / Dissertation

In this thesis we use the theory of algebraic operads to define a complete invariant of real and rational homotopy classes of maps of topological spaces and manifolds. More precisely let f,g : M -&gt; N be two smooth maps between manifolds M and N. To construct the invariant, we define a homotopy Lie structure on the space of linear maps between the homology of M and the homotopy groups of N, and a map mc from the set of based maps from M to N, to the set of Maurer-Cartan elements in the convolution algebra between the homology and homotopy. Then we show that the maps f and g are real (rational) homotopic if and only if mc(f) is gauge equivalent to mc(g), in this homotopy Lie convolution algebra. In the last part we show that in the real case, the map mc can be computed by integrating certain differential forms over certain subspaces of M. We also give a method to determine in certain cases, if the Maurer-Cartan elements mc(f) and mc(g) are gauge equivalent or not. / <p>At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 2: Manuscript. Paper 3: Manuscript.</p>

If X is a simply connected CW complex, then it has a unique (up to isomorphism) minimal Sullivan model. There is an important rational homotopy invariant, called the rational Lusternik–Schnirelmann of X, denoted cat0(X), which has an algebraic formulation in terms of the minimal Sullivan model of X. We study another such numerical invariant called the rational retraction index of X, denoted r0(X), which is defined in terms of the minimal Sullivan model of X and satisfies 0 ≤ r0(X) ≤ cat0(X). It was introduced by Cuvilliez et al. as a tool to estimate the rational Lusternik–Schnirelmann category of the total space of a fibration. In this thesis we compute the rational retraction index on a range of rationally elliptic spaces, including for example spheres, complex projective space, the biquotient Sp(1) \ Sp(3) / Sp(1) × Sp(1), the homogeneous space Sp(3)/U(3) and products of these. In particular, we focus on formal spaces and formulate a conjecture to answer a question posed in the original article of Cuvilliez et al., “If X is formal, what invariant of the algebra H∗(X;Q) is r0(X)?”

This licentiate thesis consists of two papers about topics related to representation stability for different automorphisms groups of topological spaces and manifolds. In Paper I, we study the rational homology groups of \textit{Torelli groups} of smooth, compact and orientable surfaces. The Torelli group of a smooth surface is the group of isotopy classes of orientation preserving diffeomorphisms that act trivially on the first homology group of the surface. In the paper, we study a certain class of stable homology classes, i.e. classes that exist for sufficiently large genus, and explicitly describe the image of these classes under a higher degree version of the \textit{Johnson homomorphism}, as a representation of the symplectic group. This gives a lower bound on the dimension of the stable homology of the group, as well as providing some further evidence that these homology groups satisfy representation stability for symplectic groups, in the sense of Church and Farb. In Paper II, we study pointed homotopy automorphisms of iterated wedge sums of spaces as well as boundary relative homotopy automorphisms of iterated connected sums of manifolds with a disk removed. We prove that the rational homotopy groups of these, for simply connected CW-complexes and closed manifolds respectively,  satisfy representation stability for symmetric groups, in the sense of Church and Farb.