Thesis (Ph. D.)--Lehigh University, 2002. / Includes vita. Includes bibliographical references (leaves 165-169).

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Esta dissertação expõe o estudo realizado sobre o artigo de R. Brown, citado na bibliografia, e sobre os conceitos necessários para a compreensão deste material. Entre os principais conceitos e resultados preliminares discutidos, podemos citar: topologia de espaços de funções, teoria de homotopia, espaços compactos ANR, característica de Euler de um compacto ANR, teorema de Lefschetz, espaços fibrados, e campos de caminhos. Os principais resultados discutidos na dissertação são os teoremas centrais do artigo de Brown: toda n-variedade topológica compacta admite um campo de caminhos com no máximo uma singularidade; e, uma n-variedade topológica compacta orientável admite um campo de caminhos sem singularidades se, e somente se, sua característica de Euler é zero. Discutimos também, suas respectivas consequências em teoria de ponto fixo / This essay has the purpose of exposing the studies on the paper by R. Brown, quoted on the references, and on the concepts necessary to the comprehension of it. Among the main concepts and preliminary results discussed, we can cite: topology of function spaces, homotopy theory, ANR compact spaces, Euler characteristic of a compact ANR, Lefschetz theorem, fiber spaces, and field paths. The main results discussed in the text are the central theorems presented on Brown\'s paper: every compact topological n-manifold admits a path field with at most one singularity, and a compact orientable topological n-manifold M admits a nonsingular path field if and only if the Euler characteristic of M is zero. We also discussed their consequences on fixed point theory

Finding new examples of compact simply connected spaces admitting a Riemannian metric of positive sectional curvature is a fundamental problem in differential geometry. Likewise, studying topological properties of families of manifolds is very interesting to topologists. The Eschenburg spaces combine both of those interests: they are positively curved Riemannian manifolds whose topological classification is known. There is a second family consisting of the Witten manifolds: they are the examples of compact simply connected spaces admitting Einstein metrics of positive Ricci curvature. Thirdly, there is a notion of generalized Witten manifold as well. Topologically, all three families share the same cohomology ring. This common ring structure motivates the definition of a manifold of type r, where r is the order of the fourth cohomology group. In 1991, M. Kreck and S. Stolz classified manifolds M of type r up to homeomorphism and dieomorphism using invariants s̄[subscript i](M) and s[subscript i](M), for i = 1, 2, 3. This gave rise to many new examples of nondieomorphic but homeomorphic manifolds. In this dissertation, new versions of the homeomorphism and dieomorphism classification of manifolds of type r are proven. In particular, we can replace s̄₁ and s̄₃ by the first Pontrjagin class and the self-linking number in the homeomorphism classification of spin manifolds of type r. As the formulas of the two latter invariants are in general much easier to compute, this simplifies the classification of these manifolds up to homeomorphism significantly. / Graduation date: 2012

Esta dissertação expõe o estudo realizado sobre o artigo de R. Brown, citado na bibliografia, e sobre os conceitos necessários para a compreensão deste material. Entre os principais conceitos e resultados preliminares discutidos, podemos citar: topologia de espaços de funções, teoria de homotopia, espaços compactos ANR, característica de Euler de um compacto ANR, teorema de Lefschetz, espaços fibrados, e campos de caminhos. Os principais resultados discutidos na dissertação são os teoremas centrais do artigo de Brown: toda n-variedade topológica compacta admite um campo de caminhos com no máximo uma singularidade; e, uma n-variedade topológica compacta orientável admite um campo de caminhos sem singularidades se, e somente se, sua característica de Euler é zero. Discutimos também, suas respectivas consequências em teoria de ponto fixo / This essay has the purpose of exposing the studies on the paper by R. Brown, quoted on the references, and on the concepts necessary to the comprehension of it. Among the main concepts and preliminary results discussed, we can cite: topology of function spaces, homotopy theory, ANR compact spaces, Euler characteristic of a compact ANR, Lefschetz theorem, fiber spaces, and field paths. The main results discussed in the text are the central theorems presented on Brown\'s paper: every compact topological n-manifold admits a path field with at most one singularity, and a compact orientable topological n-manifold M admits a nonsingular path field if and only if the Euler characteristic of M is zero. We also discussed their consequences on fixed point theory

Various data structures have been proposed for representing the connectivity of manifold triangle meshes. For example, the Extended Corner Table (ECT) stores V+6T references, where V and T respectively denote the vertex and triangle counts. ECT supports Random Access and Traversal (RAT) operators at Constant Amortized Time (CAT) cost. We propose two novel variations of ECT that also support RAT operations at CAT cost, but can be used to represent and process Simplicial 2-Complexes (S2Cs), which may represent star-connecting, non-orientable, and non-manifold triangulations along with dangling edges, which we call sticks. Vlist stores V+3T+3S+3(C+S-N) references, where S denotes the stick count, C denotes the number of edge-connected components and N denotes the number of star-connecting vertices. Ering stores 6T+3S+3(C+S-N) references, but has two advantages over Vlist: the Ering implementation of the operators is faster and is purely topological (i.e., it does not perform geometric queries). Vlist and Ering representations have two principal advantages over previously proposed representations for simplicial complexes: (1) Lower storage cost, at least for meshes with significantly more triangles than sticks, and (2) explicit support of side-respecting traversal operators which each walks from a corner on the face of a triangle t across an edge or a vertex of t, to a corner on a faces of a triangle or to an end of a stick that share a vertex with t, and this without ever piercing through the surface of a triangle.