The problem confronted in this thesis is that of determining direct calculations of the fundamental group of certain topological spaces.

We define an extension of the nth homotopy group which can distinguish a larger class of spaces. (E.g., a converging sequence of disjoint circles and the disjoint union of countably many circles, which have isomorphic fundamental groups, regardless of choice of basepoint.) We do this by introducing a generalization of homotopies, called component-homotopies, and defining the nth extended homotopy group to be the set of component-homotopy classes of maps from compact subsets of (0,1)n into a space, with a concatenation operation. We also introduce a method of tree-adjoinment for "connecting" disconnected metric spaces and show how this method can be used to calculate the extended homotopy groups of an arbitrary metric space.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第22232号 / 理博第4546号 / 新制||理||1653(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 新一, 教授 玉川 安騎男, 准教授 星 裕一郎 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

Submitted by JOÃO DAMASCENO DE OLIVEIRA MARQUES null (damascenomarques@ifma.edu.br) on 2017-01-05T00:21:04Z No. of bitstreams: 1 dissertacao_TFA.pdf: 664238 bytes, checksum: d5c4c0d2b31fcd154bf225146e1c3eeb (MD5) / Approved for entry into archive by Juliano Benedito Ferreira (julianoferreira@reitoria.unesp.br) on 2017-01-06T16:46:10Z (GMT) No. of bitstreams: 1 marques_jo_me_rcla.pdf: 664238 bytes, checksum: d5c4c0d2b31fcd154bf225146e1c3eeb (MD5) / Made available in DSpace on 2017-01-06T16:46:10Z (GMT). No. of bitstreams: 1 marques_jo_me_rcla.pdf: 664238 bytes, checksum: d5c4c0d2b31fcd154bf225146e1c3eeb (MD5) Previous issue date: 2016-12-20 / O objetivo principal deste trabalho é a demonstração do Teorema Fundamental da Álgebra por meio da Teoria de Homotopia. Esta teoria é uma das mais importantes da Topologia Algébrica. Para um melhor entendimento do tema faz-se uma retomada de algumas definições de Topologia Geral, em seguida estuda-se tópicos de homotopia e também o tema a eles relacionado, denominado Grupo Fundamental. De posse destas ideias demonstra-se o Teorema Fundamental da Álgebra. O texto tem como principal referência o livro [5]. / The main objective of this work is the proof of the Fundamental Theorem of Algebra through the Homotopy Theory. This theory is one of the most important in Algebraic Topology. For a better understanding of the subject one recalls some definitions of General Topology, next it is studied homotopy topics and also a related subject, namely Fundamental Group. Making use of these concepts the proof of Fundamental Theorem of Algebra is shown. The main reference for the text is the book [5].

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20160号 / 理博第4245号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 新一, 教授 岡本 久, 教授 玉川 安騎男 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

In this thesis we introduce a notion of graphs approximating actions of finitely generated groups on metric and measure spaces. We systematically investigate expansion properties of said graphs and we prove that a sequence of graphs approximating a fixed action ρ forms a family of expanders if and only if ρ is expanding in measure. This enables us to rely on a number of known results to construct numerous new families of expander (and superexpander) graphs. Proceeding in our investigation, we show that the graphs approximating an action are uniformly quasi-isometric to the level sets of the associated warped cone. The existence of such a relation between approximating graphs and warped cones has twofold advantages: on the one hand it implies that warped cones arising from actions that are expanding in measure coarsely contain families of expanders, on the other hand it provides a geometric model for the approximating graphs allowing us to study the geometry of the expander thus obtained. The rest of the work is devoted to the study of the coarse geometry of warped cones (and approximating graphs). We do so in order to prove rigidity results which allow us to prove that our construction is flexible enough to produce a number of non coarsely equivalent new families of expanders. As a by-product, we also show that some of these expanders enjoy some rather peculiar geometric properties, e.g. we can construct expanders that are coarsely simply connected.

In this dissertation we discuss various results for spaces that are wild, i.e. not locally simply connected. We first discuss periodic properties of maps from a given space to itself, similar to Sharkovskii's Theorem for interval maps. We study many non-locally connected spaces and show that some have periodic structure either identical or related to Sharkovskii's result, while others have essentially no restrictions on the periodic structure. We next consider embeddings of solenoids together with their complements in three space. We differentiate solenoid complements via both algebraic and geometric means, and show that every solenoid has an unknotted embedding with Abelian fundamental group, as well as infinitely many inequivalent knotted embeddings with non-Abelian fundamental group. We end by discussing Peano continua, particularly considering subsets where the space is or is not locally simply connected. We present reduced forms for homotopy types of Peano continua, and provide a few applications of these results.

In part one we show that for a compact, metric, locally path-connected topological space X, the shape group of X - as defined in Foundations of Shape Theory by Mardesic and Segal - is isomorphic to the inverse limit of discrete homotopy groups introduced by Conrad Plaut and Valera Berestovskii. We begin by providing the reader preliminary definitions of the fundamental group of a topological space, inverse systems and inverse limits, the Shape Category, discrete homotopy groups, and culminate by providing an isomorphism of the shape and deck groups for peano continua. In part two we develop work and provide further classification of Sharkovskii topological groups, which we call Sharkovskii Groups. We culminate in proving the fact that a locally compact Sharkovskii group must either be the real numbers if it is not compact, or a torsion-free solenoid if it is compact.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20158号 / 理博第4243号 / 新制||理||1610(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 望月 新一, 教授 岡本 久, 教授 玉川 安騎男 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

For a topological space X, the fundamental group can be topologized as a quotient of the path space with the compact-open topology. For one-dimensional or planar Peano continua, the fundamental group with this topology is a topological group if and only if it is semilocally simply connected. In particular, we demonstrate that the group operation is not continuous in this setting.