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Determination of the topological structure of an orbifold by its group of orbifold diffeomorphisms. (English) Zbl 1029.58004
Summary: We show that the topological structure of a compact, locally smooth orbifold is determined by its orbifold diffeomorphism group. Let \(\text{Diff}^r_{\text{Orb}}({\mathcal O})\) denote the \(C^r\) orbifold diffeomorphisms of an orbifold \({\mathcal O}\). Suppose that \(\Phi: \text{Diff}^r_{\text{Orb}}({\mathcal O}_1)\to \text{Diff}^r_{\text{Orb}}({\mathcal O}_2)\) is a group isomorphism between the orbifold diffeomorphism groups of two orbifolds \({\mathcal O}_1\) and \({\mathcal O}_2\). We show that \(\Phi\) is induced by a homeomorphism \(h: X_{{\mathcal O}_1}\to X_{{\mathcal O}_2}\), where \(X_{{\mathcal O}}\) denotes the underlying topological space of \({\mathcal O}\). That is, \(\Phi(f)= hfh^{-1}\) for all \(f\in \text{Diff}^r_{\text{Orb}}({\mathcal O}_1)\). Furthermore, if \(r> 0\), then \(h\) is a \(C^r\) manifold diffeomorphism when restricted to the complement of the singular set of each stratum.

58D19 Group actions and symmetry properties
57S25 Groups acting on specific manifolds
53C20 Global Riemannian geometry, including pinching
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