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Journal of Lie Theory 13 (2003), No. 2, 311--327
Copyright Heldermann Verlag 2003

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

Joseph E. Borzellino
Dept. of Mathematics, California Polytechnic State University, San Luis Obispo, CA 93407, U.S.A.

Victor Brunsden
Dept. of Mathematics and Statistics, Pennsylvania State University, 3000 Ivyside Park, Altoona, PA 16601, U.S.A.


\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.

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