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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

This Thesis consists of four independent parts. In the first part I prove that the delooping, i.e.the classifying space, of a grouplike monoid is an $H$-space if and only if its multiplication is a homotopy homomorphism. This is an extension and clarification of a result of Sugawara. Furthermore I prove that the Moore loop space functor and the construction of the classifying space induce an adjunction on the corresponding homotopy categories. In the second part I extend a result of G. Dunn, by proving that the tensorproduct $C_{n_1}\otimes\dots \otimes C_{n_j}$ of little cube operads is a topologically equivalent suboperad of $C_{n_1 \dots n_j}$. In the third part I describe operads as algebras over a certain colored operad. By application of results of Boardman and Vogt I describe a model of the homotopy category of topological operads and algebras over them, as well as a notion of lax operads, i.e. operads whose axioms are weakened up to coherent homotopies. Here the W-construction, a functorial cofibrant replacement for a topological operad, plays a central role. As one application I construct a model for the homotopy category of topological categories. C. Berger claimed to have constructed an operad structure on the permutohedras, whose associated monad is exactly the Milgram-construction of the free two-fold loop space. In the fourth part I prove that this statement is not correct.

Vertex algebras and strongly homotopy Lie algebras (SHLA) are extensively used in qunatum field theory and string theory. Recently, it was shown that a Courant algebroid can be naturally lifted to a SHLA. The 0-product in the de Rham chiral algebra has an identical formula to the Courant bracket of vector fields and 1-forms. We show that in general, a vertex algebra has an SHLA structure and that the de Rham chiral algebra has a non-zero l4 homotopy.

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)?”

We give a definition of the scanning map for configuration spaces that is equivariant under the action of the diffeomorphism group of the underlying manifold. We use this to extend the Bödigheimer-Madsen result for the stable splittings of the Borel constructions of certain mapping spaces from compact Lie group actions to all smooth actions. Moreover, we construct a stable splitting of configuration spaces which is equivariant under smooth group actions, completing a zig-zag of equivariant stable homotopy equivalences between mapping spaces and certain wedge sums of spaces. Finally we generalise these results to configuration spaces with twisted labels (labels in a fibre bundle subject to certain conditions) and extend the Bödigheimer-Madsen result to more mapping spaces.

Neste trabalho tratamos da homotopia monotônica entre trajetórias de um sistema de controle &#8721; sobre uma variedade M. Esta é uma variante apropriada da homotopia usual, na qual duas trajetórias são consideradas homotopicas se podem ser deformadas continuamente através de trajetórias. Inicialmente apresentamos alguns aspectos gerais e resultados fundamentais da teoria do controle. Em seguida, introduzimos a noção de regularidade para controles e a homotopia monotônica entre trajetórias de &#8721; geradas por essa classe de controles. Em particular, apresentamos um exemplo de um sistema que admite trajetórias que são homotópicas mas não são monotonicamente homotópicas. O objetivo principal foi entender a construção (análoga), para homotopia monotônica, de espaço de recobrimento universal. Entre outros, o conjunto &#915;(&#8721;,x) de classes de homotopia monotônica das trajetórias do sistema &#8721; a partir x &#8712; M possui uma estrutura de variedade diferenciável de mesma dimensão que a variedade M(o espaço estado). Como consequência desse resultado temos um difeomorfismo local que nos permitirá levantar &#8721; para a variedade &#915;(&#8721;,x), obtendo assim um novo sistema &#8721;^ em &#915;(&#8721;,x). A fim de compreendermos as propriedades universais de &#915;(&#8721;,x), tomamos um recobrimento &#960; : N &#8594; &#913;R(&#8721;,x) no sentido de que N é uma variedade diferenciável munida com um sistema de controle &#8721;~ e &#960; é um difeomorfismo local que leva e&#8721;~ ao &#8721;. Comparando as trajetórias de sistemas &#8721;^ e &#8721;~ construímos uma aplicação de levantamento &#402; : &#915;(&#8721;,x) &#8594; N que relaciona &#8721;^ e &#8721;~. Finalizamos este trabalho levando em conta a classe particular de sistemas simétricos, para qual os espaços de recobrimento monotônico &#915;(&#8721;,x) e topológico M~ de M coincidem. / In this work, we deal with monotonic homotopy between trajectories of a control system &#8721; on a manifold M. This is an apropriate variant of usual homotopy, where two trajectories are considered to be homotopic if they can be deformed to each other in a continuous way through trajectories. We introduce regularity for controls and consider monotonic homotopy between trajectories generated by regular controls. In particular, we present an example of a system having homotopic trajectories which are not monotonically homotopic. The main goal was to understand the construction for monotonic homotopy of the universal covering space and, in particular, the differentiable manifold structure on the set &#915;(&#8721;,x) of monotonic homotopy classes of trajectories starting at x &#8712; M. As a consequence of that result, we obtain a local diffeomorphism which permits lifting of &#8721; to another system &#8721;^ in &#915;(&#8721;,x). To consider universal properties of &#915;(&#8721;, x) we take a covering &#960; : N &#8594; &#913;R(&#8721;,x) in the sense that N is a differentiable manifold provided with a control system &#8721;~ and &#960; is a local diffeomorphism mapping &#8721;~ to &#8721;. Comparing the trajectories of &#8721;^ and &#8721;~ we construct a lifting mapping &#402; : &#915;(&#8721;,x) &#8594; N that relates &#8721;^ and &#8721;~. Finally, we take into account the particular class of symmetric systems, for which both coverings &#915;(&#8721;,x) and M~ coincide.

Sejam K, a garrafa de Klein, e K M S^ um fibrado com base S^ e fibra K. Neste trabalho estudamos o seguinte problema: dadas aplicações f, g : M M que preservam fibra sobre S^, quando o par (f, g) pode ser deformado, por uma homotopia que preserva fibra sobre S^, a um par de aplicações (f^{\'} , g^{\'} ) livre de coincidência? / Let K be the Klein bottle and let K M S be a Klein bottle bundle over S 1 . In this work we study the following question: given a pair of fiber preserving maps over S^ , when can it be deformed by a fiberwise homotopy over S into a pair of coincidence free fiber preserving maps over S^, (f^{\'} , g^{\'} ) ?

No description available.

Este trabalho é um estudo de raízes para aplicações f : S &rarr; Wn, onde S é uma superfície compacta, conexa e sem bordo e Wn é o espaço obtido pela reunião em um ponto do círculo S1 com n esferas S2 . / The propose of this work is studies the root problem for maps f : S &rarr; Wn, where S is a closed, connected, compact surface and W n is the space obtained by the one point union of circle S1 and n spheres S2.