This thesis concerns the relationship between bounded and controlled topology and in particular how these can be used to recognise which homotopy equivalences of reasonable topological spaces are homotopic to homeomorphisms. Let f : X → Y be a simplicial map of finite-dimensional locally finite simplicial complexes. Our first result is that f has contractible point inverses if and only if it is an ε- controlled homotopy equivalences for all ε > 0, if and only if f × id : X × R → Y × R is a homotopy equivalence bounded over the open cone O(Y +) of Pedersen and Weibel. The most difficult part, the passage from contractible point inverses to bounded over O(Y +) is proven using a new construction for a finite dimensional locally finite simplicial complex X, which we call the fundamental ε-subdivision cellulation X'ε. This whole approach can be generalised to algebra using geometric categories. In the second part of the thesis we again work over a finite-dimensional locally finite simplicial complex X, and use the X-controlled categories A*(X), A*(X) of Ranicki and Weiss (1990) together with the bounded categories CM(A) of Pedersen and Weibel (1989). Analogous to the barycentric subdivision of a simplicial complex, we define the algebraic barycentric subdivision of a chain complex over that simplicial complex. The main theorem of the thesis is then that a chain complex C is chain contractible in ( A*(X) A*(X) if and only if “C ¤ Z” 2 (A*(X × R) A*(X × R) is boundedly chain contractible when measured in O(X+) for a functor “ − Z” defined appropriately using algebraic subdivision. In the process we prove a squeezing result: a chain complex with a sufficiently small chain contraction has arbitrarily small chain contractions. The last part of the thesis draws some consequences for recognising homology manifolds in the homotopy types of Poincare Duality spaces. Squeezing tells us that a PL Poincare duality space with sufficiently controlled Poincare duality is necessarily a homology manifold and the main theorem tells us that a PL Poincare duality space X is a homology manifold if and only if X × R has bounded Poincare duality when measured in the open cone O(X+).

In this paper we study properties of metric spaces. We consider the collection of all nonempty closed subsets, Cl(X), of a metric space (X,d) and topologies on C.(X) induced by d. In particular, we investigate the Hausdorff topology and the Wijsman topology. Necessary and sufficient conditions are given for when a particular pseudo-metric is a metric in the Wijsman topology. The metric properties of the two topologies are compared and contrasted to show which also hold in the respective topologies. We then look at the metric space R-n, and build two residual sets. One residual set is the collection of uncountable, closed subsets of R-n and the other residual set is the collection of closed subsets of R-n having n-dimensional Lebesgue measure zero. We conclude with the intersection of these two sets being a residual set representing the collection of uncountable, closed subsets of R-n having n-dimensional Lebesgue measure zero.

Typescript. Includes vita and abstract. Includes bibliographical references (leaves 163-164) Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

<p>Let $\Theta_n$ be the group of $h$-cobordism classes of homotopy spheres, i.e. closed smooth manifolds which are homotopy equivalent to $S^n$, under connected sum. A homotopy sphere $\Sigma^n$ which is not diffeomorphic to $S^n$ is called ``exotic.'' For an oriented smooth manifold $M^n$, the {\bf inertia group} $I(M)\subset\Theta_n$ is defined as the subgroup of homotopy spheres such that $M\#\Sigma$ is orientation-preserving diffeomorphic to $M$. This thesis collects together a number of results on $I(M)$ and provides a summary of some fundamental results in Geometric Topology. The focus is on dimension $7$, since it is the smallest known dimension with exotic spheres. The thesis also provides two new results: one specifically about $7$-manifolds with certain $S^1$ actions, and the other about the effect of surgery on the homotopy inertia group $I_h(M)$.</p> / Master of Science (MSc)

The purpose of this paper is to examine selected topologies, the Vietoris topology in particular, on S(X), the collection of nonempty, closed subsets of a topological space X. Characteristics of open and closed subsets of S(X), with the Vietoris topology, are noted. The relationships between the space X and the space S(X), with the Vietoris topology, concerning the properties of countability, compactness, and connectedness and the separation properties are investigated. Additional topologies are defined on S(X), and each is compared to the Vietoris topology on S(X). Finally, topological convergence of nets of subsets of X is considered. It is found that topological convergence induces a topology on S(X), and that this topology is the Vietoris topology on S(X) when X is a compact, Hausdorff space.

One of the more important concepts in mathematics is the concept of order, that is, the description or comparison of two elements of a set in terms of one preceding or being smaller than or equal to the other. If the elements of a set, as pairs, exhibit certain order-type characteristics, the set is said to be a partially ordered set. The purpose of this paper is to investigate a special class of partially ordered sets, called lattices, and to investigate topologies induced on these lattices by specially defined order related properties called order-convergence and star-convergence.

Contact structures on 3-manifolds are 2-plane fields satisfying a set of conditions. The study of contact structures can be traced back for over two-hundred years, and has been of interest to mathematicians such as Hamilton, Jacobi, Cartan, and Darboux. In the late 1900's, the study of these structures gained momentum as the work of Eliashberg and Bennequin described subtleties in these structures that could be used to find new invariants. In particular, it was discovered that contact structures fell into two classes: tight and overtwisted. While overtwisted contact structures are relatively well understood, tight contact structures remain an area of active research. One area of active study, in particular, is the classification of tight contact structures on 3-manifolds. This began with Eliashberg, who showed that the standard contact structure in real three-dimensional space is unique, and it has been expanded on since. Some major advancements and new techniques were introduced by Kanda, Honda, Etnyre, Kazez, Matić, and others. Convex decomposition theory was one product of these explorations. This technique involves cutting a manifold along convex surfaces (i.e. surfaces arranged in a particular way in relation to the contact structure) and investigating a particular set on these cutting surfaces to say something about the original contact structure. In the cases where the cutting surfaces are fairly nice, in some sense, Honda established a correspondence between information on the cutting surfaces and the tight contact structures supported by the original manifold. In this thesis, convex surface theory is applied to the case of handlebodies with a restricted class of dividing sets. For some cases, classification is achieved, and for others, some interesting patterns arise and are investigated.

α-helical membrane proteins comprise about a quarter of all proteins in a cell and carry out a wide variety of essential cellular functions. This thesis is focused on topology analyses of bacterial membrane proteins. The topology describes the two-dimensional structural arrangement of a protein relative to the membrane. By combining large-scale experimental and bioinformatics techniques we have produced experimentally constrained topology models for the major part of the Escherichia coli membrane proteome. This represents a substantial increase in available topology information for bacterial membrane proteins. Many membrane protein structures show signs of internal duplication and approximate two-fold in-plane symmetry. We propose a step-wise pathway to explain how proteins with such internal inverted repeats have evolved. The pathway is based on the ‘positive-inside’ rule and starts with a protein that can adopt two topologies in the membrane, i.e. a “dual” topology protein. The gene encoding the dual topology protein is duplicated and eventually, through re-distribution of positively charge residues, the two resulting homologous proteins become fixed in opposite orientations in the membrane. Finally, the two proteins may fuse into one single polypeptide with an internal inverted repeat structure. Finally, we re-create the proposed step-wise evolutionary pathway in the laboratory by showing that only a small number of mutations are required in order to transform the homo-dimeric, dual topology protein EmrE into a hetero-dimeric complex composed of two oppositely oriented proteins.

In this paper, we investigate the topology convergence problem for the gossip-based Gradient overlay network. In an overlay network where each node has a local utility value, a Gradient overlay network is characterized by the properties that each node has a set of neighbors containing higher utility values, such that paths of increasing utilities emerge in the network topology. The Gradient overlay network is built using gossiping and a preference function that samples from nodes using a uniform random peer sampling service. We analyze it using tools from matrix analysis, and we prove both the necessary and sufficient conditions for convergence to a complete gradient structure, as well as estimating the convergence time. Finally, we show in simulations the potential of the Gradient overlay, by building a more efficient live-streaming peer-to-peer (P2P) system than one built using uniform random peer sampling. / <p>© 2011 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. QC 20111124</p>

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.