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Lifting surgeries to branched covering spaces

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dc.contributor.authorMontesinos Amilibia, José María
dc.contributor.authorHilden, Hugh Michael
dc.date.accessioned2023-06-21T02:02:52Z
dc.date.available2023-06-21T02:02:52Z
dc.date.issued1980
dc.description.abstractLong 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.)
dc.description.departmentDepto. de Álgebra, Geometría y Topología
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.refereedTRUE
dc.description.statuspub
dc.eprint.idhttps://eprints.ucm.es/id/eprint/17224
dc.identifier.citationA. 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.
dc.identifier.doi10.2307/199815
dc.identifier.issn0002-9947
dc.identifier.officialurlhttp://www.ams.org/journals/tran/1980-259-01/S0002-9947-1980-0561830-0/S0002-9947-1980-0561830-0.pdf
dc.identifier.relatedurlhttp://www.ams.org/
dc.identifier.urihttps://hdl.handle.net/20.500.14352/64706
dc.issue.number1
dc.journal.titleTransactions of the American Mathematical Society
dc.language.isoeng
dc.page.final161
dc.page.initial157
dc.publisherAmerican Mathematical Society
dc.rights.accessRightsrestricted access
dc.subject.cdu515.162.8
dc.subject.keywordequivariant surgery of branched coverings over the n-sphere
dc.subject.ucmTopología
dc.subject.unesco1210 Topología
dc.titleLifting surgeries to branched covering spaces
dc.typejournal article
dc.volume.number259
dspace.entity.typePublication
relation.isAuthorOfPublication7097502e-a5b0-4b03-b547-bc67cda16ae2
relation.isAuthorOfPublication.latestForDiscovery7097502e-a5b0-4b03-b547-bc67cda16ae2
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