The stratified structure of spaces of smooth orbifold mappings (Q2859243)

Let \(\mathcal{O}\) and \(\mathcal{P}\) be smooth \(C^\infty\) orbifolds without boundary and assume \(\mathcal{O}\) is compact. The authors consider the structure of the space of \(C^r\) maps from \(\mathcal{O}\) to \(\mathcal{P}\) defined with respect to four different notions of maps between orbifolds. In previous work [in: Infinite dimensional Lie groups in geometry and representation theory. Papers delivered on the occasion of the 2000 Howard fest on infinite dimensional Lie groups in geometry and representation theory, Washington, DC, USA, August 17--21, 2000. Singapore: World Scientific. 116--137 (2002; Zbl 1042.57013)] and [J. Lie Theory 18, No. 4, 979--1007 (2008; Zbl 1166.57021)], the authors have studied the notion of an \textit{(unreduced) orbifold map} and a \textit{reduced orbifold map}. Here, they introduce the additional notions of \textit{complete orbifold maps} and \textit{complete reduced orbifold maps}. These new definitions are justified by demonstrating that they can be used to define the pullback of an orbibundle.NEWLINENEWLINEThe main result of this paper is that the set \(C_{\star\mathrm{Orb}}^r(\mathcal{O},\mathcal{P})\) of \(C^r\) complete orbifold maps between \(\mathcal{O}\) and \(\mathcal{P}\) equipped with the \(C^r\) topology is a smooth \(C^\infty\) manifold and admits a local model on the space of \(C^r\) orbisections of the pullback to \(\mathcal{O}\) of the tangent orbibundle of \(\mathcal{P}\). Moreover, this separable vector space is a Banach space if \(1\leq r< \infty\) and a Fréchet space if \(r=\infty\). For each of the other three notions of orbifold maps, the space of \(C^r\) maps is described explicitly as a quotient of \(C_{\star\mathrm{Orb}}^r(\mathcal{O},\mathcal{P})\), and its structure is described. In the case of complete reduced maps, the structure is that of a smooth \(C^\infty\) (Banach or Fréchet) orbifold; in the case of (unreduced) orbifold maps, the space is a stratified space with strata modeled on smooth \(C^\infty\) (Banach or Fréchet) manifolds; and in the case of reduced orbifold maps, it is a stratified space with strata modeled on smooth \(C^\infty\) (Banach or Fréchet) orbifolds. The relation between the \(C^r\) diffeomorphism group of a compact orbifold defined using the four definitions is described, and it is demonstrated that each yields a topological group and a convenient Fréchet Lie group when \(r = \infty\).