On low-dimensional orbifolds and compact cores.

Abstract

Loosely speaking, a n-manifold is a space locally modeled on real n-space. The quotient of a n-manifold by a properly discontinuous free action is a n-manifold. The quotient of a n-manifold by a properly discontinuous, but not necessarily free, action as a n-orbifold. More generally, a n-orbifold is a space locally modeled on real n-space modulo a properly discontinuous group action. This dissertation is explicitly concerned with properties of low (i.e. 1, 2, and 3) dimensional orbifolds. Basic concepts of orbifolds are defined--continuity, suborbifold, covering map, fundamental group--then basic properties of orbifolds, including unique lifting of based paths and based path homotopies, are proved in Chapter I. Basic properties of 1-orbifolds and 2-orbifolds are investigated in Chapter II. In particular, homotopic essential embedded closed connected 1-orbifolds are shown to be parallel, and any 2-orbifold with finitely generated fundamental group is shown to have a compact suborbifold such that the inclusion map induces an isomorphism of fundamental groups. In Chapter III and IV, properties of maps of 2-orbifolds into 3-orbifolds are studied. Given a triangulation on a 3-orbifold, a method is described for making any map of a 2-orbifold into the 3-orbifold normal (or pseudo-normal) with respect to the triangulation. Then the conditions under which the map can be made normal through a sequence of isotopies are investigated. Given a metric on the 2-skeleton, the area of such a map is defined, allowing us to utilize the equivariant nature of least area maps to prove an orbifold version of Dehn's Lemma and the Loop Theorem. Finally, the Loop Theorem is employed in proving that any good 3-orbifold with finitely presented fundamental group has a suborbifold such that the inclusion map induces an isomorphism of fundamental groups.