Branched extensions of curves in compact surfaces
Cloyd L. Ezell · Transactions of the American Mathematical Society · 1980
A polymersion is a map F : M → N 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 ⩾ 1 n\, \geqslant \,1 such that F is topologically equivalent to the complex map z n {z^n} in a neighborhood about the point. (b) At each point x in the boundary of M , δ 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 ( δ M ) 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.