Orbispaces and Orbifolds from the Point of View of the Borel Construction, a new Definition

Abstract

``An orbifold is a space which is locally modeled on the quotient of a vector space by a finite group.'' This sentence is so easily said or written that more than one person has missed some of the subtleties hidden by orbifolds. Orbifolds were first introduced by Satake under the name ``V-manifold'' and rediscovered by Thurston who called them ``orbifolds''. Both of them used only faithful actions to define their orbifolds (these are the so called reduced orbifolds). This intuitive restriction is very unnatural mathematically if we want for example to study suborbifolds. Orbispaces are to topological spaces what orbifolds are to manifolds. They have been defined by Haefliger via topological groupoids. He defined the homotopy and (co)homology groups of an orbispace to be those of the classifying space of the topological groupoid. However, his definition of morphisms is complicated and technical. Ruan and Chen have tried to reformulate the definitions, but they don't seem treat the non-reduced case in a satisfactory way. Our idea is to define orbispaces and orbifolds directly via their classifying spaces. This approach allows us to easily define morphisms of orbispaces and orbifolds with all the desired good properties.