On the combinatorial-topology of branched covers of the sphere

In the 2010's, Thurston was considering the question of  understanding holomorphic mappings from the topological point of view. At that time, he introduced the balanced planar 4-regular graphs and  showed that they combinatorially characterize all cell graph Γ=f^{−1}(Σ) ⊂ S_2 where f:S_2→S_2 is an generic  orientation-preserving degree d branched covering, and Σ ⊂ S_2 is an  oriented Jordan curve passing through the critical values of f (the word generic means that the cardinality of the set of critical values  of f is 2d−2, the largest possible). In this talk we will provide a combinatorial presentation for a branched cover of the 2-sphere  generalizing completely the mentioned Thurston’s theorem. We will see that the most natural generalization of the balance condition for higher genera does not suffice for the realizability of a cell graph as a pullback graph Γ. Then, with one more imposition, we provide our mean result. After that, we will introduce and go over some operations defined on the (generalized) balanced graphs and mention some further results, if time permits.