FOLDING AND COVERING SPACES

Abstract

The project of this thesis concerns a field of mathematics called Geometric Topology, which essentially studies various structures and properties of manifolds, covering spaces and complexes.

Local isometries between Riemannian manifolds may be characterized as maps that send geodesic segments to geodesic segments of the same length .. Isometric foldings are likewise characterized by such a property, with the difference that we use piecewise geodesic segments instead of geodesic segments. The theory of isometric foldings is introduced by Robertson [19] who studies the stratification determined by the folds or the singularities and relates this structure to classical ideas of Hopf degree, and volume. Then the theory of isometric foldings, has been pushed by both Robertson and EL-Kholy [20,21] to include covering spaces and many other different aspects. Again the idea of topological folding is modeled by both of them on that of isometric folding, but in the absence of metrical structure.

The idea of cellular folding on CW-complexes is first defined by EL-Kholy and AL-Khursani [2], and various properties of this type of folding are also studied by them. The cellular folding f : M � N of surfaces M and N is one of the most interesting subjects to study, since the set of singularities of f in this case is a graph rI embedded in M .If rI is a regular graph, then the cellular folding

f is called regular folding. Regular foldings of a surface M onto a polygon

Pn with n vertices are studied by Farran, EL-Kholy and Robertson [6].