Branched covers of surfaces in 4-manifolds

Abstract

We give an algorithm for describing, as a framed link, the p-fold branched cover of (i) B 4 branched along the Seifert surface F of a link with int F pushed into int/34 (see Sect. 2); (ii) /34 union handles branched along F C B 4 (see Sect. 3); (iii) S 4 branched along a surface which, except for a trivial 2-ball, lies in S 3 (see Sect. 4); (iv) C P 2 branched along a nice surface such as a non-singular complex curve (see Sect. 5). Along the way we show how to describe the p-fold branched cover of B 4 along the ribbon disk of a ribbon link (Sect. 3), prove that the p-fold branched cover of B 4 along a Seifert surface for the unknot is trivial (Theorem 4.1, Sect. 4), show that the Mitnor fiber and various other complex surfaces can be built without 1 and 3-handles (Theorem 5.1 and corollaries), and draw the framed links for the complex surfaces, the cubic, quintic and Kummer (see Sect. 5). In Sect. I we fix conventions and notations.