V-manifold maps and the inverse mapping theorem (Q2367302)

A \(V\)-manifold (or: orbifold) is a paracompact Hausdorff space locally homeomorphic to an orbit space \(U/G\), where \(U\subset\mathbb{R}^ n\) is an open subset and \(G\) a finite group of \(C^ \infty\)-automorphisms of \(U\) whose set of fixed points has a dimension \(n-2\) at most. (Example \(M/\Gamma\) where \(M\) is a common manifold and \(\Gamma\) a properly discontinuous group of automorphisms.) The author introduces \(V\)-manifold maps (by local liftings on \(U)\), the concept of a rank of a \(V\)-mapping between \(V\)-manifolds, and studies the problem of invertibility. The concluding result: if a \(C^ \infty\)- smooth \(V\)-map \(f:V_ 1\to V_ 2\) between \(V\)-manifolds is invertible by a \(V\)-map \(g:V_ 2\to V_ 1\), then \(F\) induces an isomorphism of local \(V\)-structures on \(V_ 1\) and \(V_ 2\).