The Stratified Structure of Spaces of Smooth Orbifold Mappings

TL;DR

This paper explores the structure of various spaces of smooth orbifold maps, revealing stratified and manifold structures, and applies these findings to orbifold diffeomorphism groups, showing they form infinite-dimensional Lie groups.

Contribution

It introduces and analyzes different notions of orbifold maps, establishing their stratified and manifold structures, and characterizes orbifold diffeomorphism groups as infinite-dimensional Lie groups.

Findings

Spaces of orbifold maps have stratified structures modeled on Banach and Fréchet manifolds.

Complete orbifold maps form smooth Banach or Fréchet manifolds.

Orbifold diffeomorphism groups are infinite-dimensional Lie groups.

Abstract

We consider four notions of maps between smooth C^r orbifolds O, P with O 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 C^r maps between O and P with the C^r topology carries the structure of a smooth C^\infty Banach (r finite)/Frechet (r=infty) manifold. For the notion of complete reduced orbifold map, the corresponding set of C^r maps between O and P with the C^r topology carries the structure of a smooth C^\infty Banach (r finite)/Frechet (r=infty) orbifold. The remaining two notions carry a stratified structure: The C^r orbifold maps between O and P is locally a stratified space with strata modeled on smooth C^\infty Banach (r finite)/Frechet (r=infty) manifolds while the set of C^r reduced orbifold maps between O and P locally has the structure of a stratified space with strata modeled on smooth C^\infty Banach (r finite)/Frechet (r=infty) 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 they inherit the structure of C^\infty Banach (r finite)/Frechet (r=infty) manifolds. In fact, for r finite they are topological groups, and for r=infty they are convenient Frechet Lie groups.