The π-geography problem and the Hurwitz problem (2009)
Let d ≥ 2 be an integer and a partition of d. In this article we study the problem of for which pairs of integers (a, b) there is a branched coating F: ∑ → D2 = {z ∈ C: | z | 6 ≤ 1} that has critical values, x (∑) = −b, and such that the monodromy that is obtained when crossing the border of D2 in a positive sense belongs to the conjugation class in the symmetric group Sd determined by the π partition. Four variants of this problem are studied: i) without requiring domain connection, ii) requiring domain connection, iii) without requiring domain connection, but requiring that the coating be semi-stable, iv) requiring that the domain be related and that the coating is semi-stable. Complete solutions of the first two variants are obtained, and a partial solution of the remaining variants is obtained. It also explains how the interest in these problems arises from the study of an analogous question for functions whose domain is 4-dimensional.