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Let M be a compact Stein surface with boundary. We show that M admits infinitely many pairwise nonequivalent positive allowable Lefschetz fibrations over D 2 with bounded fibers.

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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) CP 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.

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We show that an infinite sequence of homotopy 4spheres constructed by Cappell-Shaneson are all diffeomorphic to S 4 . This generalizes previous results of Akbulut-Kirby and Gompf.

Let Kτ be a framed knot in S3 representing the 4-manifold which is obtained by attaching a 2-handle onto B4 along K with the framing τ (see (1)). Tristram (2) proved the following non-imbedding theorem for the generator of ).