Determination of the Topological Structure of an Orbifold by its Groups of Orbifold Diffeomerphisms

Josep E. Bozellino and Victor Brunsden

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

Full text of the article:

• Compressed DVI file (40 kilobytes)
• Compressed PostScript file (115 kilobytes)
• PDF file (263 kilobytes)