Made available in DSpace on 2014-06-11T19:27:09Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-11-10Bitstream added on 2014-06-13T18:30:56Z : No. of bitstreams: 1 dutra_acb_me_rcla.pdf: 707752 bytes, checksum: 003487414f094d392a97a22a4efb885b (MD5) / Neste trabalho tratamos do objeto matemático Grupo Topológico. Para este desenvolvimento, abordamos elementos básicos de Grupo e Espaço Topológico / In this work we consider the mathematical object Topological Group. For this development, we discuss the basic elements of the Group and Topological Space

This paper focuses on generalizing unoriented handlebody-links to the twisted virtual case, obtaining Reidemeister moves for handlebody-links in ambient spaces. The paper introduces a related algebraic structure known as twisted virtual bikeigebras whose axioms are motivated by the twisted virtual handlebody-link Reidemeister moves. In the research, twisted virtual bikeigebras are used to dene X-colorability for twisted virtual handlebody-links and define an integer-valued invariant of twisted virtual handlebody-links. The paper also includes example computations of the new invariants and use them to distinguish some twisted virtual handlebody-links.

This paper is a study of some of the basic properties of the metric half-space topology, a topology on a set which is derived from a metric on the set. In the first it is found that in a complete inner product space, the metric half-space topology is the same as one defined in terms of linear functionals on the space. In the second it is proven that in Rn the metric half-space topology is the same as the usual metric topology. In the third theorem it is shown that in a certain sense the nature of the metric halfspace topology generated by a norm on the space determines whether the norm is quadratic, that is to say, whether or not there exists an inner product on the space with the property that |x|^2=(x,x) for all x in the space.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, in fulfilment of the requirements for the degree of Master of Science, 2018 / Path replanning in high dimensional dynamic environments is critical to the success of interactive and reactive robotic agents. State of the art replanning algorithms typically extend sampling-based methods such as rapidly-exploring random trees (RRT) or probabilistic roadmaps (PRM). However, the speed of replanning in complex configuration spaces is relatively slow, which limits the effectiveness of robotic agents in highly dynamic environments. This thesis proposes DRM-connect, a novel generalisation of the PRM and RRT-connect algorithms, which carries out replanning in dynamic environments by executing graph searches over an underlying graph G, using lazy collision checking. If a path through the graph is not found, DRM-connect will repair the graph using a novel extension to RRT-connect, which we call PRM-connect. Additionally, we investigate using an approximate Reeb graph as the underlying graph G, which attempts to capture the underlying topology of the task manifold from prior experience. DRM-connect is tested with both a Reeb graph and na¨ıve graph in a 2-D domain and compared to RRT, while DRMconnect with a Reeb graph is tested in three 7-D domains, and compared to RRT-connect. Through simulation we show that the combination of DRM-connect and a Reeb graph typically outperforms both RRT/RRT-connect and DRM-connect with a na¨ıve graph in terms of replanning times, with minimal impact on the length of the solution path. / XL2019

<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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Knot theory is a rich topic in topology that studies the how circles can be embedded in Euclidean 3-space. One of the main questions in knot theory is how to distinguish between different types of knots efficiently. One way to approach this problem is to study knot invariants, which are properties of knots that do not change under a standard set of deformations. We give a brief overview of basic knot theory, and examine a specific knot invariant known as Khovanov homology. Khovanov homology is a homological invariant that refines the Jones polynomial, another knot invariant that assigns a Laurent polynomial to a knot. Dror Bar-Natan wrote a paper in 2002 that explains the construction of Khovanov homology and proves that it is an invariant. We follow his lead and attempt to clarify and explain his formulation in more precise detail.

Thesis advisor: Joshua E. Greene / We study ribbon cobordisms between 3-manifolds, i.e. rational homology cobordisms that admit a handle decomposition without 3-handles. We first define and study the more general notion of quasi-ribbon cobordisms, and analyze how lattice-theoretic methods may be used to obstruct the existence of a quasi-ribbon cobordism between two given 3-manifolds. Building on this and on previous work of Lisca, we then determine when there exists such a cobordism between two connected sums of lens spaces. In particular, we show that if an oriented rational homology sphere Y admitsa quasi-ribbon cobordism to a lens space, then Y must be homeomorphic to L(n, 1), up to orientation-reversal. As an application, we classify ribbon χ-concordances between connected sums of 2-bridge links. Lastly, we show that the notion of ribbon rational homology cobordisms yields a partial order on the set consisting of aspherical 3-manifolds and lens spaces, thus providing evidence towards a conjecture formulated by Daemi, Lidman, Vela-Vick and Wong. / Thesis (PhD) — Boston College, 2022. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

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In this thesis, we study the open book decompositions in high dimensional contact manifolds. We focus on the results about open book decomposition of manifolds and their relationship with contact geometry.