Branched covers of \(S^ 2\) and braid groups.

*(English)*Zbl 0865.57001This paper considers the classification of \(k\)-fold branched covers of \(S^2\) up to topological equivalence. The generic case, in which the local \(k\)-fold cover of each disk containing one branch point is made up of a 2-fold cover by one disk together with \(k - 2\) further disks which cover regularly, was dealt with in the last century by Lüroth and Clebsch, who showed that the topological type of such a cover depends only on the number of branch points and the degree \(k\) of the cover. Here the authors look at polynomial maps of degree \(k\) from \(S^2\) to itself and show that the local branching characteristics again determine the topological type for degenerate maps of small codimension, up to about \(k/4\). They also give examples in codimension \(k/2\) of covers which have the same branching characteristics but are inequivalent.

The paper includes a proof of the statement of Thom which gives necessary and sufficient conditions for a cover to correspond to a polynomial. It finishes with a complete topological classification of polynomials of degree \(\leq 6\), and also looks at some other classes of cover up to degree 4. The proofs combine graphical techniques for displaying the local branching data with use of the action of the braid groups on the set of covers. The authors note independent work by A. L. Edmonds, R. S. Kulkarni and R. E. Stong [Trans. Am. Math. Soc. 282, 773-790 (1984; Zbl 0603.57001)] and a number of related papers by other workers.

The paper includes a proof of the statement of Thom which gives necessary and sufficient conditions for a cover to correspond to a polynomial. It finishes with a complete topological classification of polynomials of degree \(\leq 6\), and also looks at some other classes of cover up to degree 4. The proofs combine graphical techniques for displaying the local branching data with use of the action of the braid groups on the set of covers. The authors note independent work by A. L. Edmonds, R. S. Kulkarni and R. E. Stong [Trans. Am. Math. Soc. 282, 773-790 (1984; Zbl 0603.57001)] and a number of related papers by other workers.

Reviewer: H.R.Morton (Liverpool)