Made available in DSpace on 2015-03-03T11:52:27Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-03-25Bitstream added on 2015-03-03T12:07:37Z : No. of bitstreams: 1 000803609.pdf: 784644 bytes, checksum: 21cd3aa175119679ab082ffb06ba43c1 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo principal deste trabalho e estudar a relação entre cohomologia de grupos e o problema de classificar grupos wallpaper, que são grupos de simetrias de certas figuras do plano chamadas padrões wallpaper. Há, a menos de equivalência, exatamente 17 grupos wallpaper, que classificamos usando teoria dos grupos e algebra linear. Dado um grupo wallpaper G, temos associado inicialmente a G um subgrupo abeliano normal T (subgrupo das translações) chamado reticulado, um grupo G0 = G=T chamado grupo ponto, uma ação de G0 sobre T (de modo que T e um ZG0-m odulo) e uma extensão do grupo G0 por T , 0 ! T ! G ! G0 ! 0. Usando o fato de que existe uma correspondência biunívoca entre o segundo grupo de cohomologia, H2(G0; T ), e o conjunto das classes de equivalência de G0 por T que dão origem a ação induzida de G0 sobre T e computando H2(G0; T ), para as várias possibilidades para G0, apresentamos um limitante superior para o número de grupos wallpaper. Para o cálculo de H2(G0; T ), para certos grupos pontos G0, utiliza-se a sequência espectral cohomológica e a sequência exata de cinco termos / The main goal of this work is to study the relation between the cohomology of groups and the problem of classifying wallpaper groups, which are symmetry groups of certain gures on the plane called wallpaper patterns. There are, up to isomorphism/equivalence, exactly 17 wallpaper groups, classi ed by using group theory and linear algebra. Given a wallpaper group G, we initially associate to G an abelian normal subgroup T (subgroup of the translations) called lattice, a group G0 = G T called point group, an action of G0 on T (in such a way that T is a ZG0-module) and an extension of the group G0 by T , 0 ?! T ?! G ?! G0 ?! 0. Using the fact that there is an one-to-one correspondence between the second cohomology of group, H2(G0; T ), and the set of equivalence classes of the extensions of G0 by T , that gives rise to the induced action of G0 on T , and computing H2(G0; T ), for the sereval possibilities for G0, we present an upper bound for the number of wallpaper groups. For the calculation of H2(G0; T ), of certain point groups G0, it is used the cohomological spectral sequence and the ve terms exact sequence

Orientador: João Peres Vieira / Banca: Alice Kimie Miwa Libardi / Banca: João Carlos Vieira Sampaio / Resumo: A Topologia Algébrica descreve a estrutura geométrica de um espaço topológico, associando a ele um sistema algébrico, geralmente um grupo ou uma sequência de grupos. À funções contínuas entre espaços topológicos correspondem homomorfismos entre grupos associados a estes espaços. Nesta dissertação, mostraremos que a homologia singular com coeficientes em Z, constituem uma teoria de homologia, baseados nos axiomas de Samuel Eilenberg e Norman Steenrod. Apresentaremos, também, resultados clássicos como a não existência de um homeomorfismo entre Rm e Rn, para m diferente de n, o teorema do ponto fixo de Brouwer e a não existência de campo vetorial não-nulo nas esferas de dimensão par / Abstract: The Algebraic Topology describes the geometrical structure of a topological space by associating an algebraic system, usually a group or a sequence of groups. To continuous functions between topological spaces correspond homomorphisms between groups associated to these spaces. In this work we will show that Singular Homology with Z-coe cients constitutes a homology theory, based on the Eilenberg-Steenrod Axioms. We also present some classical results as the nonexistence of a homeomorphism between Rm and Rn, if m ≠ n, the Brouwer's xed point theorem and the nonexistence of a non-zero vector eld in even dimension spheres / Mestre

Orientador: Maria Gorete Carreira Andrade / Banca: Ermínia de Lourdes Campello Fanti / Banca: Pedro Luiz Queiroz Pergher / Resumo: Este trabalho apresenta um estudo das teorias de cohomologia ordinária de grupos, da cohomologia de Tate e de Farrel, e algumas aplicações no contexto da Topologia Algébrica. Dentro desse contexto foram desenvolvidos, através da cohomologia de Tate, tópicos dentro da teoria de grupos com cohomologia periódica, detalhando resultados e condições necessárias e suficientes para um grupo ter essa propriedade. Como aplicação dessa teoria vimos um critério para uma função de uma esfera de homotopia em um CW-complexo ter uma (H,G)-coincidência. Também foram desenvolvidos tópicos sobre grupos satisfazendo certas condições de finitude, como por Exemplo grupos de dualidade virtual e, através da cohomologia de Farrell, apresentamos uma obstrução para grupos de dualidade virtual satisfazerem o isomorfismo de dualidade da teoria de Bieri e Eckmann / Abstract: In this work we present a study of the ordinary cohomology of groups, Tate cohomology and Farrell cohomology, and some applications in the context of Algebraic Topology. In this context we were developed topics of the theory of groups with periodic cohomology, detailing results and necessary and sufficient conditions for a group to have this property. As an application of this theory we present a criterion for a map defined in sphere homotopy in a CW-complex to have a (H,G)-coincidence. Also, we have developed some topics about groups that satisfy certain finiteness conditions, as for example, virtual duality groups. Besides, through Farrell cohomology, we present an obstruction for virtual duality groups satisfying the duality isomorphism of the theory due to Bieri and Eckmann / Mestre

Orientadores: Luiz Antonio Barrea San Martin, Lonardo Rabelo / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-22T17:26:40Z (GMT). No. of bitstreams: 1 Silva_JordanLambert_M.pdf: 1545963 bytes, checksum: 694fc3db0e24c098cadc145da4744772 (MD5) Previous issue date: 2013 / Resumo: Nesta dissertação, realizamos um estudo topológico das variedades flag reais. Encontrada a decomposição em células de Schubert de uma variedade flag, apresentamos dois invariantes topológicos sobre estas variedades: a homologia, obtida a partir do cálculo do operador fronteira da homologia celular, e a característica de Euler, cujo cálculo foi realizado para as variedades flag maximais e para as variedades grassmanianas simpléticas Lp(R2l) / Abstract: In this dissertation, we conduct a topological study of real flag manifolds. Found the Schubert cell decomposition of a flag manifold, we present two topological invariants for these manifolds: the homology, obtained from the calculation of the boundary operator of cellular homology, and the Euler characteristic, which was determinate for maximal flag manifolds and for symplectic grassmannians manifolds Lp(R2l) / Mestrado / Matematica / Mestre em Matemática

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Topological data analysis emerged as a field in the 2000s and has proven very useful for examining the shape of data sets. Using persistent homology as their main methodology researchers has succesfully applied the theory presented in this paper to study as varied subjects as robot motion, brain connectivity, network theory, finger print analysis and computer vision.  The mathematical theory behind persistent homology has traditionally required training far beyond what an average engineer would have. Therefore much theory is usually left out of presentations meant for an audience outside of a mathematics department. This paper contains a novel approach to the presentation of this theory, maintaining mathematical rigour while only using linear algebra as its building blocks.

Persistent homology is a technique of topological data analysis that seeks to understand the shape of data. We study the effectiveness of a single-layer perceptron and gradient boosted classification trees in classifying perhaps the most well-known data set in machine learning, the MNIST-Digits, or MNIST. An alternative representation is constructed, called MNIST-PD. This construction captures the topology of the digits using persistence diagrams, a product of persistent homology. We show that the models are more effective when trained on MNIST compared to MNIST-PD. Promising evidence reveals that the topology is learned by the algorithms. / Thesis / Master of Science (MSc)

Let R be a ring with an identity 1. Let A, B, and C be R-modules. The sequence A → [f above arrow] B→[g above arrow] C is exact providing f and g are R-homomorphisms and Im f =Ker g. Let 0 represent the R-module with precisely one element. An R-module J is injective if and only if for every exact sequence 0→A→ [f above arrow] B of R-modules and R-homomorphisms and every R-homomorphism g: A→J there exists an R-homomorphism h: B→J such that hf = g. This is a dual concept to that of a projective R-module. In the second chapter the idea of an injective R-module is studied quite intensively, and several different characterizations of injective · modules are proved. One of the principal results obtained is that every R-module is a submodule of an injective R-module. Further properties of injective R-modules are given in Chapter 3, including the concepts of injective dimension and an injective resolution of an R-module. Using these concepts the Shifting Theorem for injectives is proved. The basic definitions and results necessary for the development of the concept of injective for abstract categories are included in Chapter 4. An injective object is then defined in this general setting. Then the concept of an injective envelope is defined. The problems that arise, in the effort to restrict the category of topological groups to the appropriate subcategory so that the concept of an injective topological group is of interest, are investigated in Chapter 5. The development of the concept for one such restriction concludes this thesis. / Master of Science

Let G = (a,b,c,...) be a groupoid and T a topology for G with U_a denoting an open set in T that contains the element a. The topology T is admissible for G if for every a·b=c and U_c there exist U_a and U_b such that U_a·U_b c U_c. G is said to be topologically trivial if the only admissible topologies for G are the discrete and indiscrete. It is shown that finite groups are topologically trivial if and only if they are simple. It is shown that finite topologically trivial semigroups are necessarily groups. Various classes of topologically trivial groupoids are examine, and it is shown that there exist topologically trivial groupoids of every order. G is said to be right (analogously left) topologically trivial if one can find elements a·b = c in G and U_c in T such that a·U_b ⊈ U_c for all U_b in T whenever T is not trivial. G is said to be totally topologically trivial if one can find a·b = c in G and U_c in T such that a·U_b ⊈ U_c and U_a·b ⊈ U_c for all U_a and U_b in T whenever T is not trivial. Right, left, and total topologically triviality are studies for various algebraic systems. A continuity condition that always holds is exhibited as are new proofs for several old theorems. Consequences of imposing the tower topology on various algebraic systems are examined. If the proper subset I contained in the groupoid G is such that the null set, the set G, and each singleton set of the elements in G-I form the basis for an admissible topology for G, then I is called a generalized ideal in G. Properties of generalized ideals are studied at length. A function t from a groupoid G to another groupoid is called a local homomorphism if for each a and b in G there exist r and s in G such that a·b = r·s and such that t(r·s) = t(r)·t(s). Several properties of local homomorphisms are examined. / Ph. D.

This thesis covers improvements on the upperbounds for ropelength of a specific class of algebraic knots.