High distance Heegaard splittings of 3-manifolds

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dc.contributor.author Evans, Tatiana en
dc.date.accessioned 2016-01-22T03:58:59Z en
dc.date.available 2005-10-27 en
dc.date.issued 2006-08-01 en
dc.identifier.citation Topology and its Applications, 2006, 153 (14), pp. 2631 - 2647 (17) en
dc.identifier.issn 0166-8641 en
dc.identifier.uri http://hdl.handle.net/2292/28083 en
dc.description.abstract J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631–657] used the curve complex associated to the Heegaard surface of a splitting of a 3-manifold to study its complexity. He introduced the distance of a Heegaard splitting as the distance between two subsets of the curve complex associated to the handlebodies. Inspired by a construction of T. Kobayashi [T. Kobayashi, Casson–Gordon's rectangle condition of Heegaard diagrams and incompressible tori in 3-manifolds, Osaka J. Math. 25 (3) (1988) 553–573], J. Hempel [J. Hempel, 3-manifolds as viewed from the curve complex, Topology 40 (3) (2001) 631–657] proved the existence of arbitrarily high distance Heegaard splittings. In this work we explicitly define an infinite sequence of 3-manifolds {Mn} via their representative Heegaard diagrams by iterating a 2-fold Dehn twist operator. Using purely combinatorial techniques we are able to prove that the distance of the Heegaard splitting of Mn is at least n. Moreover, we show that π1(Mn) surjects onto π1(Mn−1). Hence, if we assume that M0 has nontrivial boundary then it follows that the first Betti number β1(Mn)>0 for all n⩾1. Therefore, the sequence {Mn} consists of Haken 3-manifolds for n⩾1 and hyperbolizable 3-manifolds for n⩾3. en
dc.language English en
dc.relation.ispartofseries Topology and its Applications en
dc.rights Items in ResearchSpace are protected by copyright, with all rights reserved, unless otherwise indicated. Previously published items are made available in accordance with the copyright policy of the publisher. Details obtained from http://www.sherpa.ac.uk/romeo/issn/0166-8641/ https://www.elsevier.com/about/company-information/policies/sharing en
dc.rights.uri https://researchspace.auckland.ac.nz/docs/uoa-docs/rights.htm en
dc.subject Heegaard splittings en
dc.subject Curve complex en
dc.title High distance Heegaard splittings of 3-manifolds en
dc.type Journal Article en
dc.identifier.doi 10.1016/j.topol.2005.11.003 en
pubs.issue 14 en
pubs.begin-page 2631 en
pubs.volume 153 en
pubs.author-url http://www.sciencedirect.com/science/article/pii/S0166864105002890 en
pubs.end-page 2647 en
pubs.publication-status Published en
dc.rights.accessrights http://purl.org/eprint/accessRights/RestrictedAccess en
pubs.subtype Article en
pubs.elements-id 32289 en
pubs.org-id Science en
pubs.org-id Mathematics en
dc.identifier.eissn 1879-3207 en
pubs.record-created-at-source-date 2016-01-22 en


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