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The Jordan-Schönflies theorem and the classification of surfaces. (English) Zbl 0773.57001

The first part of this easily readable paper presents a new proof of the Jordan curve theorem based on the fact that the Kuratowski graph \(K_{3,3}\) is nonplanar. The author goes on to give a new (graph- theoretic) proof of the Jordan-Schönflies theorem. This theorem is applied (again employing graph-theoretic information) to show that every connected compact surface can be triangulated. Then suitable combinatorial considerations (without using the Euler formula) enable the author to prove the classification theorem for compact connected surfaces.

MSC:

57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M15 Relations of low-dimensional topology with graph theory
57Q15 Triangulating manifolds
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