Branched extensions of curves in compact surfaces

Abstract:

A polymersion is a map $F: M \to N$ where M and N are compact surfaces, orientable or nonorientable, M a surface with boundary, where (a) At each interior point of M, there is an integer $n \geqslant 1$ such that F is topologically equivalent to the complex map ${z^n}$ in a neighborhood about the point. (b) At each point x in the boundary of M, $\delta M$, there is a neighborhood U containing x such that U is homeomorphic to F(U). A normal polymersion is one where $F(\delta M)$ is a normal set of curves in N. We are concerned with establishing a combinatorial representation for normal polymersions which map to arbitrary compact surfaces.