Covering theorems for open continuous mappings having two valences between orientable surfaces (Q1816540)

Let \({\mathcal X}\) be an open orientable surface with finite genus and finite number of boundary components, and let \({\mathcal Y}\) be a closed orientable surface. An open continuous function from \({\mathcal X}\) to \({\mathcal Y}\) is called a \((p, q)\)-map, \(0< q< p\), if it has a finite number of branch points and assumes every point in \({\mathcal Y}\) either \(p\) or \(q\) times, counting multiplicity, with possibly a finite number of exceptions. These comprise the most general class of all nontrivial functions having two valences between \({\mathcal X}\) and \({\mathcal Y}\). In this paper we generalize and prove in a unified manner almost all the earlier covering and existence results involving \((p, q)\)-maps between orientable surfaces. Our results are combinatorial in nature; they relate the branch orders and exceptional sets of \((p, q)\)-maps to the valences \(p\), \(q\) and the topological invariants of \({\mathcal X}\) and \({\mathcal Y}\). The paper ends with open questions.