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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.

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