4-manifolds, 3-fold covering spaces and ribbons.

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It is shown that a PL, orientable 4-manifold with no 3- or 4-handles is a 3-fold irregular cover of the 4-ball, branched over a ribbon 2-manifold. The author also studies 2-fold branched cyclic covers and finds examples of surfaces in S4 whose 2-fold branched covers are again S4; this gives new examples of exotic involutions on S4 [cf. C. McA. Gordon, Proc. London Math. Soc. (3) 29 (1974), 98–110]. The conjecture that any closed, orientable 4-manifold is an irregular 4-fold branched cover of S4 is reduced to studying bordism classes of irregular 4-fold covers of S3 with covering space equal to a connected sum of copies of S1×S2.
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C. Gordon, On the higher-dimensional Smith conjecture, Proc. London Math. Soc. (3) 28 (1974), 98-110. H. M. Hilden, Three-fold branched coverings of S3, Amer. J. Math. 98 (1976), 989-997. H. M. Hilden and J. M. Montesinos, A method of constructing 3-manifolds and its application to the computation of the μ-invariant, Proc. Sympos. in Pure Math., vol. 32, Amer. Math. Soc., Providence, R.I., 1977, pp. 477-485. P. Kim and J. Toilefson, Splitting the PL involutions on nonprime 3-manifolds (to appear). W. B. R. Lickorish, A representation of orientable, combinatorial 3-manifolds, Ann. of Math. (2) 76 (1962), 531-540. B. Mazur, A note on some contractible 4-manifolds, Ann. of Math. (2) 73 (1961), 221-228. J. M. Montesinos, Heegaard diagrams for closed 4-manifolds, Proc. Georgia Geometric Topology Conf., 1977. I. Berstein and A. L. Edmonds, On the construction of branched coverings of low-dimensional manifolds (preprint). I. Berstein and A. L. Edmonds, The degree and branch set of a branched covering (preprint). T. Yajima, On a characterization of knot groups of some spheres in R4, Osaka J. Math. 6 (1969), 435-446.