L’objectif de cette thèse est de démontrer que π4(S3) ≃ Z/2Z en théorie des types homotopiques. En particulier, c’est une démonstration constructive et purement homotopique. On commence par rappeler les concepts de base de la théorie des types homotopiques et on démontre quelques résultats bien connus sur les groupes d’homotopie des sphères : le calcul des groupes d’homotopie du cercle, le fait que ceux de la forme πk(Sn) avec k < n sont triviaux et la construction de la fibration de Hopf. On passe ensuite à des outils plus avancés. En particulier, on définit la construction de James, ce qui nous permetde démontrer le théorème de suspension de Freudenthal et le fait qu’il existe un entier naturel n tel que π4(S3) ≃ Z/2Z. On étudie ensuite le produit smash des sphères, on construit l’anneau de cohomologie des espaces et on introduit l’invariant de Hopf, ce qui nous permet de montrer que n est égal soit à 1, soit à 2. L’invariant de Hopf nous permet également de montrer que tous les groupes de la forme π4n−1(S2n) sont infinis. Finalement, on construit la suite exacte de Gysin, ce qui nous permet de calculer la cohomologie de CP2 et de démontrer que π4(S3) ≃ Z/2Z, et que plus généralement on a πn+1(Sn) ≃ Z/2Z pour tout n ≥ 3 / The goal of this thesis is to prove that π4(S3) ≃ Z/2Z in homotopy type theory. In particular it is a constructive and purely homotopy-theoretic proof. We first recall the basic concepts of homotopy type theory, and we prove some well-known results about the homotopy groups of spheres: the computation of the homotopy groups of the circle, the triviality of those of the form πk(Sn) with k < n, and the construction of the Hopf fibration. We then move to more advanced tools. In particular, we define the James construction which allows us to prove the Freudenthal suspension theorem and the fact that there exists a natural number n such that π4(S3) ≃ Z/nZ. Then we study the smash product of spheres, we construct the cohomology ring of a space, and we introduce the Hopf invariant, allowing us to narrow down the n to either 1 or 2. The Hopf invariant also allows us to prove that all the groups of the form π4n−1(S2n) are infinite. Finally we construct the Gysin exact sequence, allowing us to compute the cohomology of CP2 and to prove that π4(S3) ≃ Z/2Z and that more generally πn+1(Sn) ≃ Z/2Z for every n ≥ 3

We explore motivic homotopy theory over deeper bases than the spectrum of the integers: Starting from a commutative group object in a cartesian closed presentable infinity category, replacing the usual multiplicative group scheme in motivic spaces, we construct projective spaces, and show that infinite dimensional projective space is the classifying space of the group object. After passage to the stabilization, we construct a Snaith spectrum, calculate the cohomology represented by it for projective spaces and on its rationalization produce Adams operations and a splitting into summands of their eigenspaces.

This Ph.D. thesis deals with E1-ring structures on the Hermitian K-theory in the motivic setting, more precisely, the existence of such structures on the motivic spectrum representing the hermitianK-theory is proven. The presence of such structure is established through two different approaches. In both cases, we consider the category of algebraic vector bundles over a scheme, with the usual requirements to do motivic homotopy theory. This category has two natural symmetric monoidal structures given by the direct sum and the tensor product, together with a duality coming from the functor represented by the structural sheaf. The first symmetric monoidal structure is the one that we are going to group complete along this text, and we will see that the second one, the tensor product, is preserved giving rise to an E1-ring structure in the resulting spectrum. In the first case, a classic infinite loop space machine applies to the hermitian category of the category of algebraic vector bundles over a scheme. The second approach abords the construction using a new hermitian infinite loop space machine which uses the language of infinity categories. Both assemblies applied to our original category have like output a presheaf of E1-ring spectra. To get an spectrum representing the hermitian K-theory in the motivic context we need a motivic spectrum, i.e, a P1-spectrum. We use a delooping construction at the end of the text to obtain a presheaf of E1-ring P1-spectra from the two presheaves of E1-ring spectra indicated above.

In 1970, Raoul Bott published The Periodicity Theorem for the Classical Groups and Some of Its Applications, in which he uses this famous result as a guideline to present some important areas and tools of Algebraic Topology. This dissertation aims to use the path Bott presented in his article as a guideline to address certain topics on Algebraic Topology. We start this incursion developing important tools used in Homotopy Theory such as spectral sequences and Eilenberg-MacLane spaces, exploring how they can be combined to aid in computation of homotopy groups. We then study important results of Morse Theory, a tool which was in the centre of Botts proof of the Periodicity Theorem. We also develop two extensions: Morse-Bott Theory, and the applications of such results to the loopspace of a manifold. We end by giving an introduction to generalised cohomology theories and K-Theory. / Em 1970, Raoul Bott publicou o artigo The Periodicity Theorem for the Classical Groups and Some of Its Applications no qual usava esse famoso resultado como um guia para apresentar importantes áreas e ferramentas da Topologia Algébrica. O presente trabalho usa o mesmo caminho traçado por Bott em seu artigo como roteiro para explorar tópicos importantes da Topologia Algébrica. Começamos esta incursão desenvolvendo ferramentas importantes da Teoria de Homotopia como sequências espectrais e espaços de Eilenberg-MacLane, explorando como estes podem ser combinados para auxiliar em cálculos de grupos de homotopia. Passamos então a estudar resultados importantes de Teoria de Morse, uma ferramenta que estava no centro da demonstração de Bott do Teorema da Periodicidade. Desenvolvemos ainda, duas extensões: Teoria de Morse-Bott e aplicações destes resultados ao espaço de laços de uma variedade. Terminamos com uma introdução a teorias de cohomologia generalizadas e K-Teoria.

O presente trabalho é uma apresentação de aplicações de estruturas da álgebra de dimensões altas para a teoria de homotopia. Mais precisamente mostramos que existe uma equivalência entre as categorias dos cat$^n$-grupos e a dos $n$-cubos cruzados de grupos, ambas equivalentes a categoria das $n$-categorias estritas internas à categoria de grupos, e uma certa subcategoria da categoria dos $n$-cubos fibrantes, os chamados $n$-cubos de Eilenberg-MacLane. Além disso existe uma equivalência entre uma localização dessa subcategoria e a categoria homotópica dos $(n+1)$-tipos homotópicos, o que sugere a utilidade de usar as estruturas algébricas apresentadas como invariantes topológicas. O teorema central dessa teoria, o teorema generalizado de Seifert-van Kampen, diz que o funtor dos $n$-cubos de fibração aos cat$^n$-grupos usado para mostrar a equivalência mencionada preserva o colimite de certos diagramas e que nesses casos conectividade é preservada, o que permite certas computações. Apresentaremos definições das estruturas algébricas mencionadas além de como calcular certos colimites na categoria de $n$-cubos cruzados de grupos, demonstraremos os teoremas principais da teoria e mostramos como usar esses resultados para generalizar resultados clássicos da topologia algébrica como o teorema de Blakers-Massey, o teorema de Hurewicz e a fórmula de Hopf para homologia de grupos. / The present work is a presentation of applications to homotopy theory of structures in higher dimensional algebra. More precisely we show how the categories of crossed $n$-cubes of groups and of cat$^n$-groups, both equivalent to the category of strict $n$-categories internal to the category of groups, are equivalent to a subcategory of the category of fibrant $n$-cubes, namely the Eilenberg-MacLane $n$-cubes. There is also an equivalence between a localization of the category of Eilenberg-MacLane $n$-cubes and the homotopy category of homotopy $(n+1)$-types, which suggests the usefulness of the presented algebraic structures as topological invariants. The central theorem of this theory, the generalized Seifert-van Kampen theorem, states that the functor from $n$-cube of fibrations to the cat$^n$-groups used to show the aforementioned equivalence preserves the colimit of certain diagrams, and in these cases connectivity is preserved, which permits some computations. We present definitions of the relevant algebraic structures and also how to calculate certain colimits in the category of crossed $n$-cubes of groups, we demonstrate the main theorems of the theory and then we show how to generalize classical results in algebraic topology like the Blakers-Massey theorem, Hurewicz theorem and Hopf\'s formula for the homology of groups.

O presente trabalho é uma apresentação de aplicações de estruturas da álgebra de dimensões altas para a teoria de homotopia. Mais precisamente mostramos que existe uma equivalência entre as categorias dos cat$^n$-grupos e a dos $n$-cubos cruzados de grupos, ambas equivalentes a categoria das $n$-categorias estritas internas à categoria de grupos, e uma certa subcategoria da categoria dos $n$-cubos fibrantes, os chamados $n$-cubos de Eilenberg-MacLane. Além disso existe uma equivalência entre uma localização dessa subcategoria e a categoria homotópica dos $(n+1)$-tipos homotópicos, o que sugere a utilidade de usar as estruturas algébricas apresentadas como invariantes topológicas. O teorema central dessa teoria, o teorema generalizado de Seifert-van Kampen, diz que o funtor dos $n$-cubos de fibração aos cat$^n$-grupos usado para mostrar a equivalência mencionada preserva o colimite de certos diagramas e que nesses casos conectividade é preservada, o que permite certas computações. Apresentaremos definições das estruturas algébricas mencionadas além de como calcular certos colimites na categoria de $n$-cubos cruzados de grupos, demonstraremos os teoremas principais da teoria e mostramos como usar esses resultados para generalizar resultados clássicos da topologia algébrica como o teorema de Blakers-Massey, o teorema de Hurewicz e a fórmula de Hopf para homologia de grupos. / The present work is a presentation of applications to homotopy theory of structures in higher dimensional algebra. More precisely we show how the categories of crossed $n$-cubes of groups and of cat$^n$-groups, both equivalent to the category of strict $n$-categories internal to the category of groups, are equivalent to a subcategory of the category of fibrant $n$-cubes, namely the Eilenberg-MacLane $n$-cubes. There is also an equivalence between a localization of the category of Eilenberg-MacLane $n$-cubes and the homotopy category of homotopy $(n+1)$-types, which suggests the usefulness of the presented algebraic structures as topological invariants. The central theorem of this theory, the generalized Seifert-van Kampen theorem, states that the functor from $n$-cube of fibrations to the cat$^n$-groups used to show the aforementioned equivalence preserves the colimit of certain diagrams, and in these cases connectivity is preserved, which permits some computations. We present definitions of the relevant algebraic structures and also how to calculate certain colimits in the category of crossed $n$-cubes of groups, we demonstrate the main theorems of the theory and then we show how to generalize classical results in algebraic topology like the Blakers-Massey theorem, Hurewicz theorem and Hopf\'s formula for the homology of groups.

In 1970, Raoul Bott published The Periodicity Theorem for the Classical Groups and Some of Its Applications, in which he uses this famous result as a guideline to present some important areas and tools of Algebraic Topology. This dissertation aims to use the path Bott presented in his article as a guideline to address certain topics on Algebraic Topology. We start this incursion developing important tools used in Homotopy Theory such as spectral sequences and Eilenberg-MacLane spaces, exploring how they can be combined to aid in computation of homotopy groups. We then study important results of Morse Theory, a tool which was in the centre of Botts proof of the Periodicity Theorem. We also develop two extensions: Morse-Bott Theory, and the applications of such results to the loopspace of a manifold. We end by giving an introduction to generalised cohomology theories and K-Theory. / Em 1970, Raoul Bott publicou o artigo The Periodicity Theorem for the Classical Groups and Some of Its Applications no qual usava esse famoso resultado como um guia para apresentar importantes áreas e ferramentas da Topologia Algébrica. O presente trabalho usa o mesmo caminho traçado por Bott em seu artigo como roteiro para explorar tópicos importantes da Topologia Algébrica. Começamos esta incursão desenvolvendo ferramentas importantes da Teoria de Homotopia como sequências espectrais e espaços de Eilenberg-MacLane, explorando como estes podem ser combinados para auxiliar em cálculos de grupos de homotopia. Passamos então a estudar resultados importantes de Teoria de Morse, uma ferramenta que estava no centro da demonstração de Bott do Teorema da Periodicidade. Desenvolvemos ainda, duas extensões: Teoria de Morse-Bott e aplicações destes resultados ao espaço de laços de uma variedade. Terminamos com uma introdução a teorias de cohomologia generalizadas e K-Teoria.

Cette thèse de doctorat a pour sujet les modèles de la théorie homotopique des types avec l'Axiome d'Univalence introduit par Vladimir Voevodsky. L'auteur prend pour cadre de travail les définitions de type-theoretic model category, type-theoretic fibration category (cette dernière étant la notion de modèle considérée dans cette thèse) et d'univers dans une type-theoretic fibration category, définitions dues à Michael Shulman. La problématique principale de cette thèse consiste à approfondir notre compréhension de la stabilité de l'Axiome d'Univalence pour les catégories de préfaisceaux, en particulier pour les groupoïdes équipés d'une involution. / This PhD thesis deals with some new models of Homotopy Type Theory and the Univalence Axiom introduced by Vladimir Voevodsky. Our work takes place in the framework of the definitions of type-theoretic model categories, type-theoretic fibration categories (the notion of model under consideration in this thesis) and universe in a type-theoretic fibration category, definitions due to Michael Shulman. The goal of this thesis consists mainly in the exploration of the stability of the Univalence Axiom for categories of functors , especially for groupoids equipped with involutions.

Le thème de cette thèse est les différents aspects de la théorie de Borel-Moore dans le monde motivique. Classiquement, sur le corps des nombres complexes, l’homologie de Borel-Moore, aussi appelée “homologie à support compact”, possède des propriétés assez différentes comparée avec l’homologie singulière. Dans cette thèse on étudiera quelques généralisations et applications de cette théorie dans les catégories triangulées de motifs.La thèse est composée de deux parties. Dans la première partie on définit l'homologie motivique de Borel-Moore dans les catégories triangulées de motifs mixtes définies par Cisinski et Déglise et étudie ses diverses propriétés fonctorielles, tout particulièrement une fonctorialité analogue au morphisme de Gysin raffiné défini par Fulton. Ces résultats nous serviront ensuite à identifier le coeur de la structure de poids de Chow définie par Hébert et Bondarko: il se trouve que le coeur, autrement dit la catégorie des éléments de poids zéro, est équivalente à une version relative des motifs purs de Chow sur une base définie par Corti et Hanamura.Dans la deuxième partie on démontre la représentabilité de la G-théorie de Quillen, sous la reformulation de Thomason, dans un premier temps dans la catégorie A1-homotopique des schémas de Morel-Voevodsky, mais aussi dans la catégorie homotopique stable construite par Jardine. On établit une identification de celle-ci comme la théorie de Borel-Moore associée à la K-théorie algébrique, en utilisant le formalisme des six foncteurs établi par Ayoub et Cisinski-Déglise. / The theme of this thesis is different aspects of Borel-Moore theory in the world of motives. Classically, over the field of complex numbers, Borel-Moore homology, also called “homology with compact support”, has some properties quite different from singular homology. In this thesis we study some generalizations and applications of this theory in triangulated categories of motives.The thesis is composed of two parts. In the first part we define Borel-Moore motivic homology in the triangulated categories of mixed motives defined by Cisinski and Déglise and study its various functorial properties, especially a functoriality similar to the refined Gysin morphism defined by Fulton. These results are then used to identify the heart of the Chow weight structure defined by Hébert and Bondarko: it turns out that the heart, namely the category of elements of weight zero, is equivalent to a relative version of pure Chow motives over a base defined by Corti and Hanamura.In the second part we show the representability of Quillen’s G-theory, reformulated by Thomason, firstly in the A1-homotopy category of schemes of Morel-Voevodsky, but also in the stable homotopy category constructed by Jardine. We establish an identification of G-theory as the Borel-Moore theory associated to algebraic K-theory, by using the six functors formalism settled by Ayoub and Cisinski-Déglise.