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Fold maps from the sphere to the plane. (English) Zbl 1133.57019

Any stable map from a surface to the plane has an associated graph, in the case of the sphere of tree type. The trees as graphs of fold maps from the sphere to the plane and for this the sets of integers that may occur as winding numbers for the branch sets of such maps are studied. See also M. Yamamoto [Pac. J. Math. 212, No. 2, 371–376 (2004; Zbl 1056.57018)]; T. Ohmoto and F. Aicardi [Topology 45, No. 1, 27–45 (2006; Zbl 1082.57024)].

MSC:

57R45 Singularities of differentiable mappings in differential topology
57M15 Relations of low-dimensional topology with graph theory
57R65 Surgery and handlebodies
57R52 Isotopy in differential topology
58K15 Topological properties of mappings on manifolds
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