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A note on 3-fold branched coverings of S3

dc.contributor.authorMontesinos Amilibia, José María
dc.date.accessioned2023-06-21T02:06:23Z
dc.date.available2023-06-21T02:06:23Z
dc.date.issued1980-09
dc.description.abstractFor any closed orientable 3-manifold M there is a framed link (L,μ) in S3 such that M is the boundary of a 4-manifold W4(L,μ) obtained by adding 2-handles to the 4-ball along components of the framed link L. A link is symmetric if it is a union of a strongly invertible link about R1⊂R2⊂R3+ and a split link of trivial components in R3+∖R2. The author shows (Theorem 2) that there is an algorithm to obtain from a given framed link in S3 a framed symmetric link that determines the same 3-manifold. A coloured ribbon manifold (M,ω) is an immersion M in S3 with only ribbon singularities of a disjoint union of disks with handles together with a function ω from the set of components of M to the set {1,2}. Such an (M,ω) determines uniquely an oriented 4-manifold V4(M,ω) as an irregular 3-fold covering of D4, as was shown by the author [Trans. Amer. Math. Soc. 245 (1978/79), 453–467;]. Theorem 3: There is an algorithm to obtain from a framed symmetric link (L,μ) a coloured ribbon manifold (M,ω) such that W4(L,μ)≈V4(M,ω). These results yield a new proof of the theorem that each closed orientable 3-manifold is a 3-fold dihedral covering of S3, branched over a knot [cf. H. M. Hilden, Amer. J. Math. 98 (1976), no. 4, 989–997; the author, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 105, 85–94;].
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/22038
dc.identifier.doi10.1017/S0305004100057625
dc.identifier.issn0305-0041
dc.identifier.officialurlhttp://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=2083284
dc.identifier.relatedurlhttp://journals.cambridge.org/action/login
dc.identifier.urihttps://hdl.handle.net/20.500.14352/64862
dc.issue.number2
dc.journal.titleMathematical Proceedings of the Cambridge Philosophical Society
dc.language.isoeng
dc.page.final325
dc.page.initial321
dc.publisherCambridge Univ Press
dc.rights.accessRightsrestricted access
dc.subject.cdu5151.1
dc.subject.keywordLow-dimensional topology
dc.subject.ucmTopología
dc.subject.unesco1210 Topología
dc.titleA note on 3-fold branched coverings of S3
dc.typejournal article
dc.volume.number88
dcterms.referencesFox, R. H. Some problems in knot theory. Topology of 3-manifolds, ed. Fort, M. K., Prentice Hall, 1962. Hempel, J. Construction of orientable 3-manifolds. Topology of 3-manifolds, ed. Fort, M. K., Prentice Hall, 1962. Hilden, H. Every closed, orientable 3-manifold is a 3-fold branched covering space of S3. Bull. Amer. Math. Soc. 80 (1974), 1243–4. Hilden, H. Three-fold branched coverings of S3. Amer. J. Math. 98 (1976), 989–997. Hilden, H. and Montesinos, J. A method of constructing 3-manifolds and its application to the computation of the μ-invariant. Proc. of Symposia in Pure Math. A.M.S. Publications 32 (1978), 51–69. Kirby, R. A calculus for framed links in S3. Inventions Math. 45 (1978), 35–56. Lickorish, W. B. R. A representation of orientable combinatorial 3-manifolds. Ann. of Math. 76 (1962), 531–540. Lickorish, W. B. R. A finite set of generators for the homeotopy group of a 2-manifold. Proc. Cambridge Philos. Soc. 60 (1964), 769–78. Corrigendum Lickorish, W. B. R. A finite set of generators for the homeotopy group of a 2-manifold. Proc. Cambridge Philos. Soc. 62 (1966), 679–81. Montesinos, J. A representation of closed, orientable 3-manifolds as 3-fold branched coverings of S3. Bull. Amer. Math. Soc. 80 (1974), 845–6. Montesinos, J. Three-manifolds as 3-fold branched covers of S3. Quarterly J. Math. Oxford (2), 27 (1976), 85–94. Montesinos, J. 4-manifolds, 3-fold covering spaces and ribbons. Trans. Amer. Math. Soc. 245 (1978), 453–467. Wallace, A. D. Modifications and cobounding manifolds. Canad. J. Math. 12 (1960), 503–528.
dspace.entity.typePublication
relation.isAuthorOfPublication7097502e-a5b0-4b03-b547-bc67cda16ae2
relation.isAuthorOfPublication.latestForDiscovery7097502e-a5b0-4b03-b547-bc67cda16ae2

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