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Branched immersions of surfaces. (English) Zbl 0612.57020

Let M be a compact connected orientable 2-manifold with \(k>0\) boundary components and let f: Bd(M)\(\to {\mathbb{R}}^ 2\) be a stable immersion. An extension \(F: M\to {\mathbb{R}}^ 2\) is called a polymersion if locally \(F(z)=z^ n\), \(n>0\). Two such extensions are called equivalent if they differ by a diffeomorphism of M which is fixed on Bd(M). In this paper the methods of S. J. Blank are applied to the problem of extending immersions of Bd(M) into \({\mathbb{R}}^ 2.\)
Generalizing Blank’s approach for the case \(M=D^ 2\), a set of k ”words” is associated to an immersion, f, and an extension is shown to correspond to a combinatorial structure on these ”words”. Algorithms are given which answer the following questions: (1) Given an immersion of Bd(M) into the plane, is there a polymersion extension to M ? (2) Given the existence of extensions, what are the equivalence classes of the extensions ?
Each combinatorial structure can be viewed as a sequence of operations which simplify the ”words”. An extension exists if and only if the ”words” simplify to an elementary form. The collection of such structures is in one-to-one correspondence with the equivalence classes of extensions.

MSC:

57R42 Immersions in differential topology
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
57M12 Low-dimensional topology of special (e.g., branched) coverings
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