The thrust of this thesis is to describe base change BC_E/F at the level of chamber homology and K-theory for some p-adic groups, such as SL(2,F), GL(1,F) and GL(2,F). Here F is a non-archimedean local field and E is a Galois extension of F. We have had to master the representation theory of SL(2) and GL(2) including the Langlands parameters. The main result is an explicit computation of the effect of base change on the chamber homology groups, each of which is constructed from cycles. This will have an important connection with the Baum-Connes correspondence for such p-adic groups. This thesis involved the arithmetic of fields such as E and F, geometry of trees, the homology groups and the Weil group W_F.

This thesis contains a study of topological phases of matter, with a strong emphasis on symmetry as a unifying theme. We take the point of view that the "topology" in many examples of what is loosely termed "topological matter", has its origin in the symmetry data of the system in question. From the fundamental work of Wigner, we know that topology resides not only in the group of symmetries, but also in the cohomological data of projective unitary-antiunitary representations. Furthermore, recent ideas from condensed matter physics highlight the fundamental role of charge-conjugation symmetry. With these as physical motivation, we propose to study the topological features of gapped phases of free fermions through a Z₂-graded C*-algebra encoding the symmetry data of their dynamics. In particular, each combination of time reversal and charge conjugation symmetries can be associated with a Clifford algebra. K-theory is intimately related to topology, representation theory, Clifford algebras, and Z₂-gradings, so it presents itself as a powerful tool for studying gapped topological phases. Our basic strategy is to use various K</em-theoretic invariants of the symmetry algebra to classify symmetry-compatible gapped phases. The super-representation group of the algebra classifies such gapped phases, while its K-theoretic difference-group classifies the obstructions in passing between two such phases. Our approach is a noncommutative version of the twisted K-theory approach of Freed--Moore, and generalises the K-theoretic classification first suggested by Kitaev. It has the advantage of conceptual simplicity in its uniform treatment of all symmetries. Physically, it encompasses phenomena which require noncommutative algebras in their description; mathematically, it clarifies and provides rigour to the meaning of "homotopic phases", and easily explains the salient features of Kitaev's Periodic Table.

Several basic topics from Algebraic Topology, including fundamental group and universal covering space are shown. The hyperbolic plane is defined, including its metric and show what the "straight" lines are in the plane and what the isometries are on the plane. A hyperbolic surface is defined, and shows that the two hole torus is a hyperbolic surface, the hyperbolic plane is a universal cover for any hyperbolic surface, and the quotient space of the universal cover of a surface to the group of automorphisms on the covering space is equivalent to the original surface.

No description available.

Un réseau de capteurs sans fil résulte du déploiement d'un ensemble de petites unités autonomes interagissant via un réseau construit grâce à leur module de communication qui observent leur environnement par des capteurs pour ensuite traiter et/ou sauvegarder cette information via leur capacité calculatoire et de stockage. La couverture est la seule représentation disponible aux réseaux de capteurs de l'espace physique environnant. Par conséquent, il est essentiel de pouvoir qualifier et quantifier sa qualité notamment concernant la présence de trous. Nos travaux utilisent la topologie algébrique pour répondre à ces problèmes. Plus précisément, nous définissons dans un premier temps une notion de trou de couverture d'un champ scalaire qui mesure la qualité de l'estimation par le réseau de capteurs sans pour autant connaître la position des capteurs. Cela permet d'utiliser l'homologie simpliciale pour déterminer la qualité de la couverture globale et accessoirement de mettre en veille certains capteurs surnuméraires tout en garantissant la couverture. Puis, afin de rendre le résultat précédent facilement calculable par un réseau de capteurs grâce à une distribution du calcul qui supporte en plus le passage à l'échelle, nous utilisons la théorie de Morse discrète pour faire le calcul des groupes d'homologie nécessaires à notre application précédente. Enfin, cette dernière approche est rendue suffisamment souple pour permettre le suivi temporel des modifications de la couverture de manière délocalisée. Cela permet non seulement de suivre la qualité de la couverture lorsque l'environnement se modifie mais aussi de proposer un schéma distribué de mise en veille des capteurs afin d'augmenter la durée de vie du réseau de capteurs tout en garantissant une couverture suffisante. / A wireless sensor network consists of a set of small autonomous units that interact via a network built by their communication modules. They observe their environment by their sensors and then they manage this information according to their computational capacity and storage. The coverage is the only representation available to the sensor network of its environment. Therefore, it is essential to quantify the quality of coverage especially related to the presence of holes. Our work uses algebraic topology to solve these problems. We first define a notion of the coverage hole in a scalar field, which measures the quality of the estimation by the sensor network without knowing the positions of the sensors. It allows the simplicial homology tool to determine the quality of the overall coverage and put certain redundant sensors into sleeping mode with the guarantee of the coverage. Then, to make the previous result easier to compute by a sensor network, the discrete Morse theory is used. It allows a distributed computation of the previous homology groups while supporting scalability necessary in sensor networks domain. Finally, one flexible approach that allows time varying tracking which allows a coverage is proposed in a distributed way. When the environment changes, this approach can not only guarantee the capability of monitoring of coverage quality, but also proposes a scheme to send to sleep the redundant sensors in order to increase the lifetime of the sensor network with adequate coverage.

Topological order is a new paradigm for quantum phases of matter developed to explain phase transitions which do not fit the symmetry breaking scheme for classifying phases of matter. They are characterized by patterns of entanglement that lead to topologically depended ground state degeneracy and anyonic excitations. One common approach for studying such phases in two-dimensional systems is through exactly solvable lattice Hamiltonian models such as quantum double models and String-Net models. The former can be understood as the Hamiltonian formulation of lattice gauge theories and, as such, it is defined by a finite gauge group. However, not much is known about topological phases in tridimensional systems. Motivated by this we develop a new class of three-dimensional exactly solvable models which go beyond quantum double models by using finite crossed modules instead of gauge groups. This approach relies on a lattice implementation of 2-gauge theory to obtain models with a richer topological structure. We construct the Hamiltonian model explicitly and provide a rigorous proof that the ground state degeneracy is a topological invariant and that the ground states can only be characterized with nonlocal order parameters. / Ordem topológica é um novo paradigma para fases quânticas da matéria desenvolvido para explicar transições de fase que não se encaixam no esquema de classificação de fases da matéria por quebra de simetria. Estas fases são caracterizadas por padrões de emaranhamento que levam a uma degenerescência de estado fundamental topológica e a excitações anyonicas. Uma abordagem comum para o estudo de tais fases em sistemas bidimensionais é através de modelos Hamiltonianos exatamente solúveis de rede como os modelos duplos quânticos e modelos de String-Nets. O primeiro pode ser entendido como a formulação Hamiltoniana de teorias de gauge na rede e, desta maneira, é definido por um group de gauge finito. Entretanto, pouco é conhecido a respeito de fases topológicas em sistemas tridimensionais. Motivado por isso nós desenvolvemos uma nova classe de modelos tridimensionais exatamente solúveis que vai alem de modelos duplos quânticos pelo uso de módulos cruzados finitos no lugar de grupos de gauge. Esta abordagem se baseia numa implementação em redes de teoria de 2-gauge para obter modelos com uma estrutura topológica mais rica. Nós construímos o modelos Hamiltoniano explicitamente e fornecemos uma demonstração rigorosa de que a degenerescência de estado fundamental é um invariante topológico e que os estados fundamentais só podem ser caracterizados por parâmetros de ordem não locais.

No description available.

Betrachtet man das kartesische Produkt X × X eines topologischen Raumes X mit sich selbst, so kann auf diesem Objekt insbesondere die Involution betrachtet werden, die die Koordinaten vertauscht, die also (x,y) auf (y,x) abbildet. Das sogenannte 'Symmetrische Quadrieren' in Čech-Homologie mit Z/2-coefficients wurde von Schick et al. 2007 als Abbildung von der k-ten Čech-Homologiegruppe eines Raumes X in die 2k-te Čech-Homologiegruppe von X × X modulu der oben genannten Involution definiert. Es stellt sich heraus, dass diese Konstruktion entscheidend ist für den Beweis eines parametrisierten Borsuk-Ulam-Theorems.Das Symmetrische Quadrieren kann zu einer Abbildung in Bordismus verallgemeinert werden, was der Hauptgegenstand dieser Dissertation ist. Genauer gesagt werden wir zeigen, dass es eine wohldefinierte, natürliche Abbildung von der k-ten singulären Bordismusgruppe von X in die 2k-te Bordismusgruppe von X × X modulu der obigen Involution gibt.Insbesondere ist dieses Quadrieren wirklich eine Verallgemeinerung der Konstruktion in Čech-Homologie, denn es ist vertauschbar mit dem Übergang von Bordismus zu Homologie via dem Fundamentalklassenhomomorphismus. Auf dem Weg zu diesem Resultat wird das Konzept des Čech-Bordismus als Kombination aus Bordismus und Čech-Homologie zunächst definiert und dann mit Čech-Homologie verglichen.

The prehistory of Algebraic Topology dates back to Euler, Riemann and Betti, who started the idea of attaching various invariants to a topological space. With his simplicial (co)homology theory, Poincaré was the first to give an instance of what in modern terms we would call a contravariant functor H° from the category of (sufficiently nice) topological spaces to the category of cyclic complexes of abelian groups.

Orientador: Alice Kimie Miwa Libardi / Banca: Pedro Luiz Queiroz Pergher / Banca: Denise de Mattos / Resumo: O objetivo principal deste trabalho é apresentar um estudo sobre Homologia de Espaços Fibrados, baseado no livro Elements of Homotopy Theory de G.W.Whitehead. O conceito de fibração apareceu em torno de 1930 e pode ser visto como uma extensão da teoria de fibrados. Existe uma sequência exata longa que relaciona os grupos de homotopia dos espaços base, total e da fibra de uma fibração. Porém, relacionar os grupos de homologia desses espaços é uma tarefa mais complicada. O caso geral é feito utilizando sequências espectrais. Porém, há casos particulares em que podemos obter relações sem utilizar a maquinaria das sequências espectrais / Abstract: The main goal of this work is to present a study on Homology of Fibre Spaces, based on the book of G.W. Whitehead: "Elements of Homotopy Theory". The concept of fibration appeared around 1930 and can be seen as an extension of the theory of bundles. There is a long exact sequence that relates the homotopy groups of the total, base and fiber spaces of a fibration. However, relating the homology groups of such spaces is more complicated. The general case is obtained using spectral sequences. Nevertheless there are particular cases where one can obtain such relations without the need of the machinery of spectral sequences / Mestre