Orientador: Elíris Cristina Rizziolli / Banca: Aldício José Miranda / Banca: João Peres Vieira / Resumo: Esse trabalho trata de elementos da Topologia Algébrica, a qual tem como fundamental aplicação abordar questões acerca de Espaços Topológicos sob o ponto de vista algébrico. Uma das questões é tentar responder se dois espaços topológicos X e Y são homeomorfos. Neste sentido, o grupo fundamental é uma ferramenta algébrica útil por se tratar de um invariante topológico. Além disso, apresentamos o Teorema de van Kampen do ponto de vista da Linguagem de Categorias e Funtores / Abstract: This work treats of elements of the Algebraic Topology, which has as fundamental application to approach subjects concerning Topological Spaces under the algebraic point of view. One of the subjects is to try to answer if two topological spaces X and Y are homeomorphics. In this sense, the fundamental group is an useful algebraic tool for treating of an topological invariant. In addition, we presented the van Kampen's Theorem of the point of view of the language of Categories and Functors / Mestre

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In the paper \cite{FHS12}, the authors announce the discovery of an invariant for Grothendieck toposes which they call the isotropy group of a topos. Roughly speaking, the isotropy group of a topos carries algebraic data in a way reminiscent of how the subobject classifier carries spatial data. Much as we like to compute invariants of spaces in algebraic topology, we would like to have tools to calculate invariants of toposes in category theory. More precisely, we wish to be in possession of theorems which tell us how to go about computing (higher) isotropy groups of various toposes. As it turns out, computation of isotropy groups in toposes can often be reduced to questions at the level of small categories and it is therefore interesting to try and see how isotropy behaves with respect to standard constructions on categories. We aim to provide a summary of progress made towards this goal, including results on various commutation properties of higher isotropy quotients with colimits and the way isotropy quotients interact with categories collaged together via certain nice kinds of profunctors. The latter should be thought of as an analogy for the Seifert-van Kampen theorem, which allows computation of fundamental groups of spaces in terms of fundamental groups of smaller subspaces.

<p>Secondary homological stability is a recently discovered stability pattern for the homology of a sequence of spaces exhibiting homological stability in a range where homological stability does not hold. We prove secondary homological stability for the homology of the unordered configuration spaces of a connected manifold. The main difficulty is the case that the manifold is compact because there are no obvious maps inducing stability and the homology eventually is periodic instead of stable. We resolve this issue by constructing a new chain-level stabilization map for configuration spaces.</p>

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Thesis (M.Sc. (Mathematics)) -- University of Limpopo, 2022 / Our discussion starts with the study of convergence and clustering of filters initiated in pointfree setting by Hong, and then characterize compact and almost compact frames in terms of these filters. We consider the strict extension and show that tQL is a zerodimensional compact frame, where Q denotes the set of filters in L. Furthermore, we study the notion of general filters introduced by Banaschewski and characterize compact frames and almost compact frames using them. For filter selections, we consider F−compact and strongly F−compact frames and show that lax retracts of strongly F−compact frames are also strongly F−compact. We study further the ideals Rs(L) and RK(L) of the ring of realvalued continuous functions on L, RL. We show that Rs(L) and RK(L) are improper ideals of RL if and only if L is compact. We consider also fixed ideals of RL and showthat L is compact if and only if every ideal of RL is fixed if and only if every maximalideal of RL is fixed. Of interest, we consider the class of isocompact locales, which is larger that the class of compact frames. We show that isocompactness is preserved by nearly perfect localic surjections. We study perfect compactifications and show that the Stone-Cˇech compactifications and Freudenthal compactifications of rim-compact frames are perfect. We close the discussion with a small section on Z−closed frames and show that a basically disconnected compact frame is Z−closed.

The dissertation uses a number of mathematical formula including de Rham cohomology with complex coefficients to state and prove extension of Brieskorn's Lemma theorem.