Introduction to Orbifolds

Orbifolds lie at the intersection of many different areas of mathematics, including algebraic and differential geometry, topology, algebra and string theory. Orbifolds were first introduced into topology and differential geometry by Satake [6], who called them V-manifolds. Satake described them as topological spaces generalizing smooth manifolds and generalized concepts such as de Rham cohomology and the Gauss-Bonnet theorem to orbifolds. In the late 1970s, orbifolds were use by Thurston in his geometrization program for three-manifolds. It was Thurston who changed the name from V-manifold to orbifold. In 1985, with the work of Dixon, Harvey, Vafa and Witten on conformal field theory [7], the interest on orbifolds dramatically increased, due to their role in string theory, even though orbifolds were already very important objects in mathematics. As Thurston mentions in [1, p. 297], it is often more effective to study the quotient manifold of a group acting freely and properly discontinuously on a space rather than to limit one’s image to the group action alone. In the same spirit, it is often more effective to study the quotient spaces of groups acting properly discontinuously, but not necessarily freely, on a topological space rather than to limit one’s image to the action alone. Since a way to construct orbifolds is by taking the quotient of a manifold by some properly discontinuous group action, as we will see in the next sections, the study of orbifolds often simplify the analysis of more complicated structures, such as three-manifolds, for example. From the ideas discussed in the paragraph above, we can think of orbifolds as a space with isolated singularities, that is, a space that looks like a quotient manifold of a group acting on a space, together with some additional information about the action of the group on points where the action is not free. For instance, if I consider the quotient space of the disk D by the action of the group of rotations of order 3 around the center of the disk, our orbifold would be the quotient space together with information telling me that the group of rotations acts as the cyclic group of order 3 at the origin. In this paper, we introduce the basics of the topology of orbifolds, talk about their fundamental groups and state an orbifold version of van Kampen’s theorem. With this machinery, we show that PSL2(Z) is isomorphic to the free product of a cyclic group of order two and another of order three.