Branched immersions of surfaces (Q1088212)

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.