Browsing by Author "Schlagbauer, H"
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- ItemOpen AccessAn approach to coincidence theory through universal covering spaces(1973) Harvey, Duncan Reginald Arthur; Schlagbauer, HThe close relationship between the theory of fixed points and the theory of coincidences of maps is well known. This presentation is aimed at recording one of the less well documented approaches to fixed point theory as extended to the more general situation of coincidences. The approach referred to is that by way of the Universal Covering Spaces. The existing theory of coincidences is geometrically well realised in this setting and after some consideration, the necessary extensions and generalizations of the techniques as utilized in fixed point theory lead to an appealing conceptual notion of "essentiality of coincidence classes". Many hints have been made in the literature (see [1] and "On the sharpness of the Δ₂ and Δ₁ Nielsen Numbers" by Robin Brooks, J.Reine Angew. Math. 259, (1973), 101-108.) that lifts of mappings and the theory of fibres and related topics lend themselves to coincidence theory. It is the intention of this presentation to follow some of the basic properties through this approach and to show, wherever it is thought desirable, the ties between this and two of the existing approaches - for example, in the definition of the Nielsen Number, which is fundamental to both fixed point theory and coincidence theory.
- ItemOpen AccessCompactifications, subordinations and uniformities(1968) Salbany, Sergio de Ornelas; Schlagbauer, HThis thesis relates hausdorff compactifications of spaces and other structures on the space. Each chapter starts with an introduction, which describes its content, and ends with a collection of notes, where the references relevant to the chapter are given. Most of the thesis is self-contained. The standard reference throughout is: General Topology, by Kelley. The notation is as in Kelley. We often use the abbreviations - s. t. for such that; iff for if and only if; i.e. for that is. This thesis arose from an attempt to prove the essential steps in the construction of vX, described by Aleksandrov in his survey Some Results in the Theory of Topological Spaces, Obtained Within the Last Twenty-Five Years.
- ItemOpen AccessThe Delta-Nielsen number in products(1973) Mordant, Ian; Schlagbauer, HIn 1967 Robert F. Brown derived a formula which relates the Nielsen number N(f) of a fibre map f to the Nielsen numbers N(f),(fb), where f,fb are induced by f. This work is concerned to prove an analogous result for the Δ-Nielsen number, N(f,g,Δ). In Chapter I we introduce the set of coincidences of two maps f,g: X->Γ,f(f,g) = {xϵX: f(x)=g(x)}. We partition this set into equivalence classes by means of the equivalence relation of fixed end-point homotopy and then study some of the geometry of the equivalence classes. We then proceed to introduce the Δ-Nielsen number N(f,g,Δ) by means of an index, which we show satisfies the axioms of Brooks [1969] for a coincidence index. Thereafter we show N(f,g,Δ) to be a homotopy invariant. In Chapter II we introduce the class of fibre spaces. By restricting ourselves to fibre spaces which are products of closed, finitely triangulable manifolds, we derive an analogous formula for coincidences as Brown has for fixed points. Some suggestions for a complete analogue conclude the work.
- ItemOpen AccessRhythmomachia : a propaedeutic game of the middle ages(1984) Coughtrie, Margareta Emma; Du Ry, Charles; Schlagbauer, HBibliography: leaves 226-246.