Determination of the Topological Structure of an Orbifold by its Group of Orbifold Diffeomorphisms

J. E. Borzellino ; V. Brunsden

Journal of Lie theory, Tome 13 (2003) no. 2, pp. 311-327.

Résumé

\def\Diff{\hbox{Diff}} \def\OrB{\hbox{\footnotesize{Orb}}} We show that the topological structure of a compact, locally smooth orbifold is determined by its orbifold diffeomorphism group. Let $\Diff^r_{\OrB}(\cal{O})$ denote the $C^r$ orbifold diffeomorphisms of an orbifold $\cal{O}$. Suppose that $\Phi\colon\Diff^r_{\OrB} ({\cal{O}}_1) \to \Diff^r_{\OrB}({\cal{O}}_2)$ is a group isomorphism between the the orbifold diffeomorphism groups of two orbifolds ${\cal{O}}_1$ and ${\cal{O}}_2$. We show that $\Phi$ is induced by a homeomorphism $h\colon X_{{\cal{O}}_1} \to X_{{\cal{O}}_2}$, where $X_{\cal{O}}$ denotes the underlying topological space of $\cal{O}$. That is, $\Phi(f)=h f h^{-1}$ for all $f\in \Diff^r_{\OrB}({\cal{O}}_1)$. Furthermore, if $r > 0$, then $h$ is a $C^r$ manifold diffeomorphism when restricted to the complement of the singular set of each stratum.