The classical Steiner Problem may be stated: Given n points [formula omitted] in the Euclidean plane, to construct the shortest tree(s) (i.e. undirected, connected, circuit free graph(s)) whose vertices include [formula omitted]. The problem is generalised by considering sets in a metric space rather than points in E² and also by minimising a more general graph function than length, thus yielding a large class of network minimisation problems which have a wide variety of practical applications, The thesis is concerned with the following aspects of these problems. 1. Existence and uniqueness or multiplicity of solutions. 2. The structure of solutions and demonstration that minimising trees of various problems share common properties. 3. Solvability of problems by Euclidean constructions or by other geometrical methods. / Science, Faculty of / Mathematics, Department of / Graduate

Cyclic surgery on satellite knots in S³ is classified and a necessary condition is given for a knot in S³ to admit a nontrivial cyclic surgery with slope m/l, \m\ > 1. A complete classification of cyclic group actions on the Poincaré sphere with 1-dimensional fixed point sets is obtained. It is proved that the following knots have property I, i.e. the fundamental group of the manifold obtained by Dehn surgery on such a knot cannot be the binary icosahedral group I₁₂₀, the fundamental group of the Poincaré homology 3-sphere: nontrefoil torus knots, satellite knots, nontrefoil generalized double knots, periodic knots with some possible specific exceptions, amphicheiral strongly invertible knots, certain families of pretzel knots. Further the Poincaré sphere cannot be obtained by Dehn surgery on slice knots and a certain family of knots formed by band-connect sums. It is proved that if a nonsufficiently large hyperbolic knot in S³ admits two nontrivial cyclic Dehn surgeries then there is at least one nonintegral boundary slope for the knot. There are examples of such knots. Thus nonintegral boundary slopes exist. / Science, Faculty of / Mathematics, Department of / Graduate

The aim of this dissertation at the outset is to give a survey of obstruction theories after Steenrod and to describe the various techniques employed by different researchers, the intricate perhaps subtle relation from one technique to another. Owing to the difficulty in computing higher co-homology operations, one is led naturally to K-theory and the Eilenberg-Moore spectral sequence. However, these and other recent developments especially those in the study of stable Postnikov systems go beyond the intention of this modest survey. / Science, Faculty of / Mathematics, Department of / Graduate

Given a topological space (X,Ʊ), let H(X,Ʊ), be the class of all homeomorphisms of (XƱ ) onto itself. This paper is devoted to study the following problem posed by Everett and Ulam [1], [11] in 1948. When and how a new topology Ʋ can be constructed on X such that H(X,Ʊ) = H(X,Ʋ), i.e., these two topological spaces have exactly the same class of homeomorphisms. Some of the results obtained are original, and other results agree essentially with the work done previously by Yu-Lee Lee [5], [6], [7], [8], [9]. / Science, Faculty of / Mathematics, Department of / Graduate

The study of invariant means on spaces of functions associated with a group or semigroup has been the interest of many mathematicians since von Neumann's work on invariant measures appeared in 1929. In recent years, many important properties of locally compact groups have been found to depend on the existence of an invariant mean on a suitable translation-invariant space of functions on the group. In this thesis, we deal mostly with invariant means on the space L∞G) of bounded measurable functions on a locally compact group G. Several characterisations of the existence of an invariant mean on L∞G) are given. Among other results, we prove the remarkable theorem that L∞(G) has a left invariant mean if and only if G is topologically right stationary, an analogue of a recent result for semigroups by T. Mitchell. However our approach is entirely different. / Science, Faculty of / Mathematics, Department of / Graduate

In this paper we are concerned with the problem of finding 'limits' of inverse (or projective) systems of measure spaces (for a definition of these see e.g. Choksi: Inverse Limits of Measure, Spaces, Proc. London Math. Soc. 8, 1958). Our basic limit measure, ῦ, is placed on the Cartesian product of the spaces instead of on the inverse limit set, L. As a result we obtain an existence theorem for this measure with fewer conditions on the system than are usually needed. We also investigate the existence of a limit measure on L by restricting our measure ῦ to L. This enables us to generalize known results and to explain some of the difficulties encountered by the standard inverse limit measure. In particular we show that the product topology may be too fine to allow the limit measure to have good topological properties' (e.g. to be Radón). Another topology which is related to the product structure is introduced and we show that limit measures which are Radón w.r.t. this topology can be obtained for a wide class of inverse systems of measure spaces. / Science, Faculty of / Mathematics, Department of / Graduate

Two topologies (Z,L) and (Z,L1) for the family of the non-degenerate convex curves in the projective plane are considered, where (Z,L) is the topology from the Lane's neighborhood system and (Z,L1) is the topology from the parabolic neighborhood system. It is shown that the definition of convexity in the affine plane can be extended to the projective plane so that the Blaschke selection theorem remains true for the projective convex sets. With the help of this theorem, the topological space (Z,L) is compactified by adding Lane's compactifying elements. Furthermore, it is shown that (Z,L) is metrizable but (Z,L1) is not metrizable. The Lane's topology (X,L), as a subspace of (Z,L) for the non-degenerate conics, is both metrizable and separable. A subspace (X,τ) of (Z,L1) is studied which is metrizable but not separable. / Science, Faculty of / Mathematics, Department of / Graduate

Given a set X , let P(X) be the collection of all subsets of X . A nonempty sub-collection u, of P(X) is called a generalized topology for X and the ordered pair (X, u) is called a generalized topological space or an abstract space or simply a space. Elements of u are said to be u-open and their complements are said to be u-closed. We define u-closure, u-limit point, ….. and so on in the natural way. Most of the basic notions in point set topology are defined analogously. It is expected that many important results in point set topology will not be carried over and a number of interesting properties will be lost or weakened. Nevertheless, some of them will still hold true despite the absence of the finite intersection axiom and the arbitrary union axiom for the collection of subsets. The primary objective of this thesis is to investigate which theorems in point set topology still remain valid in our more general setting. A secondary objective is to provide some counterexamples showing certain basic results in point set topology turn out to be false in the setting. It should be noted that other basic notions which are not discussed here at all can be defined similarly. However, in order to attain desirable and interesting conclusions, additional conditions must be imposed. / Science, Faculty of / Mathematics, Department of / Graduate

We introduce open quasitoric manifolds and their functorial properties, including complex bundle maps of their stable tangent bundles, and relate these new spaces to the standard constructions of toric topology: quasitoric manifolds, moment angle manifolds and polyhedral products. We extend the domain of these constructions to countably infinite simplicial complexes, clarifying and generalising constructions of Davis and Januszkiewicz. In particular we describe final spaces in the categories of open quasitoric manifolds and quasitoric spaces, as well as in the categories of characteristic pairs and dicharacteristic pairs. We show how quasitoric manifolds can be constructed smoothly as pullbacks of the final spaces QT(n) for n >= 1, and how stably complex structure also arises this way. We calculate the integral cohomology of quasitoric spaces over Cohen-Macaulay simplicial complexes, including the final spaces QT(n) as a special case. We describe a procedure for calculating the Chern numbers of a quasitoric manifold M and, relating this to our cohomology calculations, show how it may be interpreted in terms of the simplicial homology of H(n), the simplicial complex underlying QT(n).

The graded cohomology groups of a cartesian product of two cellular spaces are expressible in terms of the cohomology groups of the factors. This relationship is given by the (split) short exact Runneth sequence. However the multiplicative structure on the cohomology of a cartesian product can in general not be derived by solely referring to the Runneth formula. In this thesis we explicitly exhibit the cup product structure on a cartesian product of two (standard) lens spaces. This result is obtained by analyzing the Runneth sequence and by making use of the particular geometry of the spaces involved. / Science, Faculty of / Mathematics, Department of / Graduate