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Lifting surgeries to branched covering spaces



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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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Official URL: http://www.ams.org/journals/tran/1980-259-01/S0002-9947-1980-0561830-0/S0002-9947-1980-0561830-0.pdf



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
Deposited On:28 Nov 2012 09:25
Last Modified:12 Dec 2018 15:14

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