Realizability and exceptionality of candidate surface branched covers: methods and results

Let f : Σ̃ d:1 −→ Π Σ denote a branched cover between closed, connected, and orientable surfaces, where d is the global degree, Π = {Π1, . . . ,Πn}, n is the number of branching points, and Πi is the partition of d given by the local degrees over the i-th branching point. If l(Π) is the total length of Π, then the Riemmann-Hurwitz formula asserts that χ(Σ̃) − l(Π) = d · (χ(Σ) − n). A candidate branched cover is a symbol Σ̃ d:1 99K Π Σ satisfying the same condition, and it is called realizable if there exists a corresponding f : Σ̃ d:1 −→ Π Σ. The problem of determining which candidate covers are realizable is very old and still partially unsolved. In this paper we will review five different techniques employed in recent years to attack it, and we will state the main results obtained using them. Each technique will be exemplified through a proof that the candidate cover S2 4:1 99K99K99K (2,2),(2,2),(3,1) S2 is exceptional, namely non-realizable. We will also state some results for the non-orientable version of the problem, none of which is due to us.