Branched Covers of $S^2$ and Braid Groups

A.G.Khovanskii and Smilka Zdravkovska

We study branched covers of $S^2$, i.e. meromorphic functions on Riemann
surfaces, up to topological equivalence.  Luroth (1871) and Clebsch (1873)
proved that the topological type of a branched cover of $S^2$ in generic
position is determined by the number of branch points (critical values)
and the number of leaves of the cover.  We give some conditions for the
local characteristics around the critical values of a nongeneric branched
cover to determine its topological type.

Specifically,

(1) in the case of polynomial maps $S^2 \rightarrow S^2$ we prove that the
topological type of the cover is uniquely determined by its local
characteristics not only in the generic case, but also for all degenerate
maps of sufficiently small codimension. Namely, we prove the uniqueness of
the topological type for all degeneracies up to codimension about $k/4$,
where $k$ is the degree of the polynomial, and give examples to show the
nonuniqueness in codimension about $k/2$.

(2) We give a simple combinatorial proof showing that the topological type
of a map with only one degenerate critical value is determined by its
local branching characteristics. A topological proof of this was given by
Natanzon (1988).

(3) We give the topological classification of polynomials of degree $\leq
6$ and compute the number of topological classes of some special 3- and
4-fold covers of $S^2$.

(4) We prove the statement of Thom (1965) that each set of local
characteristics satisfying some simple (necessary) conditions actually
does correspond to a polynomial.