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Lifting surgeries to branched covering spaces

Montesinos Amilibia, José María and Hilden, Hugh Michael (1980) Lifting surgeries to branched covering spaces. Transactions of the American Mathematical Society, 259 (1). pp. 157-161. ISSN 0002-9947

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Abstract

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.)

Item Type:Article
Uncontrolled Keywords:equivariant surgery of branched coverings over the n-sphere
Subjects:Sciences > Mathematics > Topology
ID Code:17224
References:

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.

Deposited On:28 Nov 2012 09:25
Last Modified:07 Feb 2014 09:44

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