THE STRATIFIED STRUCTURE OF SPACES OF SMOOTH ORBIFOLD MAPPINGS

Abstract

We consider four notions of maps between smooth C∞ orbifolds [Formula: see text], [Formula: see text] with [Formula: see text] compact (without boundary). We show that one of these notions is natural and necessary in order to uniquely define the notion of orbibundle pullback. For the notion of complete orbifold map, we show that the corresponding set of Cr maps between [Formula: see text] and [Formula: see text] with the Cr topology carries the structure of a smooth C∞ Banach (r finite)/Fréchet (r = ∞) manifold. For the notion of complete reduced orbifold map, the corresponding set of Cr maps between [Formula: see text] and [Formula: see text] with the Cr topology carries the structure of a smooth C∞ Banach (r finite)/Fréchet (r = ∞) orbifold. The remaining two notions carry a stratified structure: The Cr orbifold maps between [Formula: see text] and [Formula: see text] is locally a stratified space with strata modeled on smooth C∞ Banach (r finite)/Fréchet (r = ∞) manifolds while the set of Cr reduced orbifold maps between [Formula: see text] and [Formula: see text] locally has the structure of a stratified space with strata modeled on smooth C∞ Banach (r finite)/Fréchet (r = ∞) orbifolds. Furthermore, we give the explicit relationship between these notions of orbifold map. Applying our results to the special case of orbifold diffeomorphism groups, we show that they inherit the structure of C∞ Banach (r finite)/Fréchet (r = ∞) manifolds. In fact, for r finite they are topological groups, and for r = ∞ they are convenient Fréchet Lie groups.