Orbifold structure gives a natural stratification by open manifolds on its underlying space, where one stratum corresponds to a set of singular points of the same type.
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Laurent Siebenmann in his thesis found an invariant similar to Wall's that gives an obstruction to an open manifold being the interior of a compact manifold with boundary.
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By construction, it contains X _ { \ mu > \ epsilon } as a dense open submanifold; essentially, it compactifies this open manifold with the symplectic quotient
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