An approach to coincidence theory through universal covering spaces - Harvey, Duncan Reginald Arthur (Department of Mathematics and Applied Mathematics, 2016)
The 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.