Topological classification of branched coverings of the two-dimensional sphere

The following questions are considered: 1) which coverings of the two-sphere besides the simple ones are determined by the obvious invariants, the number of branch points, and their types, 2) in which cases can one add to the collection of obvious invariants simple combinatorial invariants such that the collection obtained determines the covering of the sphere up to homeomorphism. It is shown that in some cases the Arf-invariant and signature introduced by the author are such additional invariants. To prove the results one develops a reduction of the problem of classification of branched coverings of the sphere to a combinatorial problem due to Hurwitz.