UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A study of orbifolds Milley, Peter Lawrence

Abstract

This thesis is a study of the theory of orbifolds and their applications in low-dimensional topology and knot theory. Orbifolds are a generalization of manifolds, and provide a larger, richer context for many of the concepts of manifold theory, such as covering spaces, fibre bundles, and geometric structures. Orbifolds are intimately connected with both the theory of Seifert fibrations and with knot theory, both of which are connected to the theory and classification of three-dimensional manifolds. Orbifolds also provide a new way to visualize group actions on manifolds, specifically actions which are not free. In chapter one we motivate the discussion with the history of orbifolds, and then we define orbifolds and certain key related terms. We extend the theory of orbifolds in chapter two to encompass many of the concepts of manifold theory, such as fibre bundles, covering spaces, and orbifold geometry. We also see in chapter two the proof that every orbifold with a geometric structure is covered by a manifold, a result which does not have an analogue in manifold theory. In chapter three we study and classify compact two-dimensional orbifolds, and show how to construct hyperbolic geometric structures for a vast majority of such orbifolds. We examine the connections between orbifolds and Seifert fibrations in chapter four. We pass on to three dimensions in chapter five. In that chapter we not only study the local structure of three-dimensional orbifolds, we also study polyhedral orbifolds and examine the consequences of Andreev's theorem. We also look at ways of constructing orbifolds from Dehn surgery diagrams in chapter five. Finally in chapter six we discuss more advanced topics, such as the state of the Geometrization Theorem for orbifolds, as well as orbifold differential geometry and orbifold topological invariants including extensions of the fundamental group and the homology groups.

Item Media

Item Citations and Data

Rights

For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.