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Bott\'s periodicity theorem from the algebraic topology viewpoint / O teorema da periodicidade de Bott sob o olhar da topologia algébricaBonatto, Luciana Basualdo 23 August 2017 (has links)
In 1970, Raoul Bott published The Periodicity Theorem for the Classical Groups and Some of Its Applications, in which he uses this famous result as a guideline to present some important areas and tools of Algebraic Topology. This dissertation aims to use the path Bott presented in his article as a guideline to address certain topics on Algebraic Topology. We start this incursion developing important tools used in Homotopy Theory such as spectral sequences and Eilenberg-MacLane spaces, exploring how they can be combined to aid in computation of homotopy groups. We then study important results of Morse Theory, a tool which was in the centre of Botts proof of the Periodicity Theorem. We also develop two extensions: Morse-Bott Theory, and the applications of such results to the loopspace of a manifold. We end by giving an introduction to generalised cohomology theories and K-Theory. / Em 1970, Raoul Bott publicou o artigo The Periodicity Theorem for the Classical Groups and Some of Its Applications no qual usava esse famoso resultado como um guia para apresentar importantes áreas e ferramentas da Topologia Algébrica. O presente trabalho usa o mesmo caminho traçado por Bott em seu artigo como roteiro para explorar tópicos importantes da Topologia Algébrica. Começamos esta incursão desenvolvendo ferramentas importantes da Teoria de Homotopia como sequências espectrais e espaços de Eilenberg-MacLane, explorando como estes podem ser combinados para auxiliar em cálculos de grupos de homotopia. Passamos então a estudar resultados importantes de Teoria de Morse, uma ferramenta que estava no centro da demonstração de Bott do Teorema da Periodicidade. Desenvolvemos ainda, duas extensões: Teoria de Morse-Bott e aplicações destes resultados ao espaço de laços de uma variedade. Terminamos com uma introdução a teorias de cohomologia generalizadas e K-Teoria.
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Koliha–Drazin invertibles form a regularitySmit, Joukje Anneke 10 1900 (has links)
The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms
are satisfied. However, there arise a number of spectra, usually defined for a single element
of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and
V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was
to describe the underlying set of elements on which the spectrum is defined. The axioms of a
regularity provide important consequences. We prove that the set of Koliha-Drazin invertible
elements, which includes the Drazin invertible elements, forms a regularity. The properties of
the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)
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Bott\'s periodicity theorem from the algebraic topology viewpoint / O teorema da periodicidade de Bott sob o olhar da topologia algébricaLuciana Basualdo Bonatto 23 August 2017 (has links)
In 1970, Raoul Bott published The Periodicity Theorem for the Classical Groups and Some of Its Applications, in which he uses this famous result as a guideline to present some important areas and tools of Algebraic Topology. This dissertation aims to use the path Bott presented in his article as a guideline to address certain topics on Algebraic Topology. We start this incursion developing important tools used in Homotopy Theory such as spectral sequences and Eilenberg-MacLane spaces, exploring how they can be combined to aid in computation of homotopy groups. We then study important results of Morse Theory, a tool which was in the centre of Botts proof of the Periodicity Theorem. We also develop two extensions: Morse-Bott Theory, and the applications of such results to the loopspace of a manifold. We end by giving an introduction to generalised cohomology theories and K-Theory. / Em 1970, Raoul Bott publicou o artigo The Periodicity Theorem for the Classical Groups and Some of Its Applications no qual usava esse famoso resultado como um guia para apresentar importantes áreas e ferramentas da Topologia Algébrica. O presente trabalho usa o mesmo caminho traçado por Bott em seu artigo como roteiro para explorar tópicos importantes da Topologia Algébrica. Começamos esta incursão desenvolvendo ferramentas importantes da Teoria de Homotopia como sequências espectrais e espaços de Eilenberg-MacLane, explorando como estes podem ser combinados para auxiliar em cálculos de grupos de homotopia. Passamos então a estudar resultados importantes de Teoria de Morse, uma ferramenta que estava no centro da demonstração de Bott do Teorema da Periodicidade. Desenvolvemos ainda, duas extensões: Teoria de Morse-Bott e aplicações destes resultados ao espaço de laços de uma variedade. Terminamos com uma introdução a teorias de cohomologia generalizadas e K-Teoria.
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Ordered spaces of continuous functions and bitopological spacesNailana, Koena Rufus 11 1900 (has links)
This thesis is divided into two parts: Ordered spaces of Continuous Functions and
the algebras associated with the topology of pointwise convergence of the associated
construct, and Strictly completely regular bitopological spaces.
The Motivation for part of the first part (Chapters 2, 3 and 4) comes from the
recent study of function spaces for bitopological spaces in [44] and [45]. In these
papers we see a clear generalisation of classical results in function spaces ( [14] and
[55]) to bi-topological spaces. The well known definitions of the pointwise topology and
the compact open topology in function spaces are generalized to bitopological spaces,
and then familiar results such as Arens' theorem are generalised. We will use the same
approach in chapters 2, 3 and 4 to formulate analogous definitions in the setting of
ordered spaces. Well known results, including Arens' theorem, are also generalised
to ordered spaces. In these chapters we will also compare function spaces in the
category of topological spaces and continuous functions, the category of bi topological
spaces and bicontinuous functions, and the category of ordered topological spaces and
continuous order-preserving functions. This work has resulted in the publication of
[30] and [31].
Continuing our study of Function Spaces, we oonsider in Chapters 5 and 6 some
Categorical aspects of the construction, motivated by a series of papers which includes
[39], [40], [41] and [50]. In these papers the Eilenberg-Moore Category of algebras of
the monad induced by the Hom-functor on the categories of sets and categories of
topological spaces are classified. Instead of looking at the whole product topology we
will restrict ourselves to the pointwise topology and give examples of the EilenbergMoore Algebras arising from this restriction. We first start by way of motivation, with
the discussion of the monad when the range space is the real line with the usual topology.
We then restrict our range space to the two point Sierpinski space, with the aim
of discovering a topological analogue of the well known characterization of Frames as
the Eilenberg-Moore Category of algebras associated with the Hom-F\mctor of maps
into the Sierpinski space [11]. In this case the order structure features prominently, resulting in the category Frames with a special property called "balanced" and Frame
homomorphisms as the Eilenberg-Moore category of M-algebras. This has resulted
in [34].
The Motivation for the second part comes from [20] and [15]. In [20], J. D. Lawson
introduced the notion of strict complete regularity in ordered spaces. A detailed study
of this notion was done by H-P. A. Kiinzi in [15]. We shall introduce an analogous
notion for bitopological spaces, and then shall also compare the two notions in the categories
of bi topological spaces and bicontinuous functions, and of ordered topological
spaces and continuous order-preserving functions via the natural functors considered
in the previous chapters. We further study the Stone-Cech bicompactification and
Stone-Cech ordered compactification in the two categories. This has resulted in [32] and [33] / Mathematical Sciences / D. Phil. (Mathematics)
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