Branched extensions of curves in orientable surfaces

Ezell, Cloyd L.; Marx, Morris L.

doi:10.1090/S0002-9947-1980-0567094-6

Branched extensions of curves in orientable surfaces
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by Cloyd L. Ezell and Morris L. Marx PDF

Trans. Amer. Math. Soc. 259 (1980), 515-532 Request permission

Abstract:

Given a set of regular curves ${f_1} , \ldots , {f_\rho }$ in an orientable surface N, we are concerned with the existence and structure of all sense-preserving maps $F: M \to N$ where (a) M is a bordered orientable surface with $\rho$ boundary components ${K_1},\ldots , {K_\rho }$, (b) $F|{K_i} = {f_i}, i = 1, \ldots , \rho$, (c) at each interior point of M, there is an integer n such that F is locally topologically equivalent to the complex map $w = {z^n}$.

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