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Note on a theorem of Munkres. (English) Zbl 1227.57032

Summary: We prove that given a \({\mathcal C}^\infty\) Riemannian manifold with boundary, having a finite number of compact boundary components, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result extends to \({\mathcal C}^1\) manifolds and to embedded \(PL\) manifolds of dimensions 2, 3 and 4. We employ these results to prove that manifolds of the types above admit quasimeromorphic mappings onto \(\widehat{{\mathbb R}^n}\). As an application, we prove the existence of \(G\)-automorphic quasimeromorphic mappings, where \(G\) is a Kleinian group acting on \(\mathbb H^n\).

MSC:

57R05 Triangulating
57M60 Group actions on manifolds and cell complexes in low dimensions
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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