Lifting surgeries to branched covering spaces

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Hilden, Hugh Michael
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American Mathematical Society
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Long ago J. W. Alexander showed that any closed, orientable, triangulated n-manifold can be expressed as a branched covering of the n-sphere [Bull. Amer. Math. Soc. 26 (1919/20), 370–372; Jbuch 47, 529]. In general, the branch set is not a manifold and no useful information is given about the degree of the branched covering. When n=3, however, he did indicate that the branch set could be arranged to be a link. Much more recently, the first author [Amer. J. Math. 98 (1976), no. 4, 989–997], U. Hirsch [Math. Z. 140 (1974), 203–230] and the second author [Quart. J. Math. Ser. (2) 27 (1976), no. 105, 85–94] showed that when n=3 the branched covering can be constructed to have degree 3 and a knot as branch set. Of course, these branched coverings are highly irregular. The authors here address similar questions in higher dimensions. Starting with a branched covering Mn→Sn, the authors give some technical, sufficient conditions for a manifold obtained from Mn by a single surgery to be a branched covering of Sn of the same degree and with a branch set easily described in terms of the initial branch set. The nicest corollary of the general technique is that if Mn→Sn is a branched covering of degree d, then there is a branched covering Mn×Sk→Sn+k of degree d+1. The new branch set is an orientable and/or locally flat submanifold if and only if the original branch set is. In particular, the n-torus is an n-fold branched covering of the n-sphere, branched along a locally flat, orientable submanifold. (For known cohomological reasons, n is the smallest possible degree of such a branched covering.)
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A. Edmonds, Extending a branched covering over a handle (preprint). cf. I. Berstein and A. Edmonds, On the construction of branched coverings of low-dimensional manifolds, Trans. Amer. Math. Soc. 247 (1979), 87-124. I. Berstein and A. Edmonds, The degree and branch set of a branched covering, Invent. Math. (to appear). J. Montesinos, Sobre la conjetura de Poincaré y los recubridores ramificados sobre un nudo, Thesis, Universidad Complutense, Madrid, Spain, 1971. J. Montesinos, Three-manifolds as 3-fold branched covers of S3, Quart. J. Math. Oxford Ser. (2) 27 (1976), 85-94. J. Montesinos, 4-manifolds, 3-fold branched covering spaces and ribbons, Trans. Amer. Math. Soc. 245 (1978), 453-467.