Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, August 1997. / Includes bibliographical references. Also available on the Internet.

<p>The work presented in this dissertation includes the study of cohomology and cohomological operations within the framework of Persistence. Although Persistence was originally defined for homology, recent research has developed persistent approaches to other algebraic topology invariants. The work in this document extends the field of persistence to include cohomology classes, cohomology operations and characteristic classes. </p><p>By starting with presenting a combinatorial formula to compute the Stiefel-Whitney homology class, we set up the groundwork for Persistent Characteristic Classes. To discuss persistence for the more general cohomology classes, we construct an algorithm that allows us to find the Poincar'{e} Dual to a homology class. Then, we develop two algorithms that compute persistent cohomology, the general case and one for a specific cohomology class. We follow this with defining and composing an algorithm for extended persistent cohomology. </p><p>In addition, we construct an algorithm for determining when a cohomology class is decomposible and compose it in the context of persistence. Lastly, we provide a proof for a concise formula for the first Steenrod Square of a given cohomology class and then develop an algorithm to determine when a cohomology class is a Steenrod Square of a lower dimensional cohomology class.</p> / Dissertation

Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 122-126).

Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-10-08Bitstream added on 2014-06-13T20:47:46Z : No. of bitstreams: 1 ruy_ac_me_rcla.pdf: 1015949 bytes, checksum: 61d6b1a36c30772dee7e55eba23514a7 (MD5) / 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 / 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

Orientador: Maria Gorete Carreira Andrade / Banca: Ermínia de Lourdes Campello Fanti / Banca: Denise de Mattos / Resumo: O teorema clássico de Borsuk-Ulam afirma que se f : Sn ����! Rn e uma aplicação contínua, então existe um ponto x na esfera tal que f(x) = f(����x). Desde a publicação, diversas generalizações desse resultado têm sido abordadas. Algumas generalizações consistem em substituir o domínio (Sn;A), onde A e a involução antipodal, por outros pares (X; T) de involuções livres, ou o contradomínio Rn por espaços topológicos mais gerais Y . Nesse caso, dizemos que ((X; T); Y ) satisfaz a propriedade de Borsuk-Ulam se dada uma aplicação contínua f : X ����! Y , existe um ponto x em X tal que f(x) = f(T(x)). Neste trabalho, detalhamos a demonstração de um resultado de classificação apresentado por Gonçalves em [6], que fornece condições necessárias e suficientes para que uma superfície fechada satisfaça a propriedade de Borsuk-Ulam. Mostramos também uma prova detalhada de um resultado apresentado por Desideri, Pergher e Vendrúsculo em [3], que estabele um critério algébrico para que um espaço topológico qualquer satisfaça a propriedade de Borsuk-Ulam / Abstract: The classic Borsuk-Ulam theorem states that if f : Sn ����! Rn is a continuous map, then there exists a point x in the sphere such that f(x) = f(����x). Since the publication, many generalizations of that result have been studied. Some generalizations consist in replacing either the domain (Sn;A), where A is the antipodal involution, by other free involution pair (X; T), or the target space Rn by more general topological spaces Y . In that case, we say that ((X; T); Y ) satisfies the Borsuk-Ulam property if given any continuous map f : X ����! Y , there exists a point x in X such that f(x) = f(T(x)). In this work, we detail the proof of a classification result presented by Gonçalves in [6], that provides necessary and suficient conditions for a closed surface satisfy the Borsuk-Ulam property. We also show a detailed proof of a result presented by, Desideri, Pergher and Vendrúsculo in [3], that establishes an algebraic criterion for any topological space satisfy the Borsuk-Ulam property / Mestre

Orientador: Alice Kimie Miwa Libardi / Banca: Daniel Vendrúscolo / Banca: Eliris Cristina Rizziolli / Resumo: O objetivo dessa dissertação é apresentar parte do artigo [2] de Gunnar Carlsson, onde se discute a adaptação de métodos da teoria usual de homologia para problemas de reconhecimento topológico de padrões em conjuntos de dados. Esta adaptação conduz aos conceitos de homologia de persistência e de barcodes. Atualmente, várias aplicações são obtidas com o uso deste método. Apresentaremos alguns casos onde a homologia de persistência é usada, ilustrando diferentes modos em que podem ser aplicados. Descreveremos, também baseado no artigo de Carlsson, um novo método para estudar a persistência de características topológicas através de uma família de conjuntos de dados, chamado persistência zig-zag . Este método generaliza a teoria de homologia de persistência e chama atenção de situações que não são cobertas pela outra teoria. Além disso, são apresentadas algumas aplicações dessa ferramenta para a obtenção de informações de alguns conjuntos de dados / Abstract: The main goal of this work is to present a part of the Gunnar Carlsson paper [2], where the adaptation of the theory of usual homology to topological pattern recognition problems in point cloud data sets is discussed. This adaptation leads to the concepts of persistence homology and barcodes. Several applications have been obtained using this method. We will present some cases where persistence homology is used, illustrating different ways in which the method can be applied. We will describe,also basedin the Carlsson's paper, a new method to study the persistence of topological features through point cloud data sets, called zig-zag persistence. This method generalizes the homology persistent theory and we will pay attention to situations that are not covered by the other theory. In addition, some applications of this tool are presented to obtain information from some data sets / Mestre

x, 72 p. A print copy of this thesis is available through the UO Libraries. Search the library catalog for the location and call number. / The theory of equivariant homology and cohomology was first created by Bredon in his 1967 paper and has since been developed and generalized by May, Lewis, Costenoble, and a host of others. However, there has been a notable lack of computations done. In this paper, a version of the Serre spectral sequence of a fibration is developed for RO ( G )-graded equivariant cohomology of G -spaces for finite groups G . This spectral sequence is then used to compute cohomology of projective bundles and certain loop spaces. In addition, the cohomology of Rep( G )-complexes, with appropriate coefficients, is shown to always be free. As an application, the cohomology of real projective spaces and some Grassmann manifolds are computed, with an eye towards developing a theory of equivariant characteristic classes. / Adviser: Daniel Dugger

Let n > 1. We prove a homological stability theorem for the diffeomorphism groups of (4n+1)-dimensional manifolds, with respect to forming the connected sum with (2n-1)-connected, (4n+1)-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold M on the linking form associated to the homology groups of M. In order to study this action we construct a geometric model for the linking form using the intersections of embedded and immersed Z/k-manifolds. In addition to our main homological stability theorem, we prove several results regarding disjunction for embeddings and immersions of Z/k-manifolds that could be of independent interest.

We construct categories of spectra for two model categories. The first is the category of small categories with the canonical model structure, and the second is the category of directed graphs with the Bisson-Tsemo model structure. In both cases, the category of spectra is homotopically trivial. This implies that the Goodwillie derivatives of the identity functor in each category, if they exist, are weakly equivalent to the zero spectrum. Finally, we give an infinite family of model structures on the category of small categories.

Made available in DSpace on 2015-04-09T12:28:27Z (GMT). No. of bitstreams: 0 Previous issue date: 2014-02-20Bitstream added on 2015-04-09T12:47:32Z : No. of bitstreams: 1 000811736.pdf: 442400 bytes, checksum: 037b5d630eff63eb854ef35fecab8412 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O teorema clássico de Borsuk-Ulam afirma que se f : Sn ! Rn e uma aplicação contínua, então existe um ponto x na esfera tal que f(x) = f(x). Desde a publicação, diversas generalizações desse resultado têm sido abordadas. Algumas generalizações consistem em substituir o domínio (Sn;A), onde A e a involução antipodal, por outros pares (X; T) de involuções livres, ou o contradomínio Rn por espaços topológicos mais gerais Y . Nesse caso, dizemos que ((X; T); Y ) satisfaz a propriedade de Borsuk-Ulam se dada uma aplicação contínua f : X ! Y , existe um ponto x em X tal que f(x) = f(T(x)). Neste trabalho, detalhamos a demonstração de um resultado de classificação apresentado por Gonçalves em [6], que fornece condições necessárias e suficientes para que uma superfície fechada satisfaça a propriedade de Borsuk-Ulam. Mostramos também uma prova detalhada de um resultado apresentado por Desideri, Pergher e Vendrúsculo em [3], que estabele um critério algébrico para que um espaço topológico qualquer satisfaça a propriedade de Borsuk-Ulam / The classic Borsuk-Ulam theorem states that if f : Sn ! Rn is a continuous map, then there exists a point x in the sphere such that f(x) = f(x). Since the publication, many generalizations of that result have been studied. Some generalizations consist in replacing either the domain (Sn;A), where A is the antipodal involution, by other free involution pair (X; T), or the target space Rn by more general topological spaces Y . In that case, we say that ((X; T); Y ) satisfies the Borsuk-Ulam property if given any continuous map f : X ! Y , there exists a point x in X such that f(x) = f(T(x)). In this work, we detail the proof of a classification result presented by Gonçalves in [6], that provides necessary and suficient conditions for a closed surface satisfy the Borsuk-Ulam property. We also show a detailed proof of a result presented by, Desideri, Pergher and Vendrúsculo in [3], that establishes an algebraic criterion for any topological space satisfy the Borsuk-Ulam property