Determination of the Topological Structure of an Orbifold by its Group of Orbifold Diffeomorphisms
Journal of Lie Theory, Volume 13 (2003) no. 2, pp. 311-327
\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.
@article{JOLT_2003_13_2_a0,
author = {J. E. Borzellino and V. Brunsden},
title = {Determination of the {Topological} {Structure} of an {Orbifold} by its {Group} of {Orbifold} {Diffeomorphisms}},
journal = {Journal of Lie Theory},
pages = {311--327},
year = {2003},
volume = {13},
number = {2},
doi = {10.5802/jolt.306},
zbl = {1029.58004},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.306/}
}
TY - JOUR AU - J. E. Borzellino AU - V. Brunsden TI - Determination of the Topological Structure of an Orbifold by its Group of Orbifold Diffeomorphisms JO - Journal of Lie Theory PY - 2003 SP - 311 EP - 327 VL - 13 IS - 2 UR - https://jolt.centre-mersenne.org/articles/10.5802/jolt.306/ DO - 10.5802/jolt.306 ID - JOLT_2003_13_2_a0 ER -
%0 Journal Article %A J. E. Borzellino %A V. Brunsden %T Determination of the Topological Structure of an Orbifold by its Group of Orbifold Diffeomorphisms %J Journal of Lie Theory %D 2003 %P 311-327 %V 13 %N 2 %U https://jolt.centre-mersenne.org/articles/10.5802/jolt.306/ %R 10.5802/jolt.306 %F JOLT_2003_13_2_a0
J. E. Borzellino; V. Brunsden. Determination of the Topological Structure of an Orbifold by its Group of Orbifold Diffeomorphisms. Journal of Lie Theory, Volume 13 (2003) no. 2, pp. 311-327. doi: 10.5802/jolt.306
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