The π-geography problem and the Hurwitz problem

Let d > 2 be an integer and let π be a partition of d. This article aims to determine for which pairs of integers (a, b) there exists a branched cover F : Σ → D 2 = {z ∈ C : |z| 6 1} with χ(Σ) = ？b and having a critical values, such that the monodromy obtained when traversing the boundary of D 2 once and positively belongs to the conjugacy class in the symmetric group Sd determined by π. Four variants of this question are studied: i) without requiring the connectedness of the domain, ii) requiring the connectedness of the domain, iii) without requiring the connectedness of the domain but requiring the semistability of the map, iv) requiring the connectedness of the domain and the semistability of the map. Complete solutions are obtained of the ？rst two variants, and partial solutions are obtained of the remaining variants. The article also explains how these questions arise when analogous questions for maps whose domain is four dimensional are studied