Heegaard Splittings and Surgery on 2- and 3-Manifolds

Heegaard Splittings and Surgery on 2- and 3-Manifolds | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Systematic Review Heegaard Splittings and Surgery on 2- and 3-Manifolds Christopher de la Viesca This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7435624/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract A Heegaard splitting of a 3-manifold is a representation of it as the union of two handlebodies with the same boundary. Each splitting is defined by an attaching homeomorphism between the boundaries of these two handle-bodies. We discuss surgery on surfaces (2-manifolds) to explain why specifying precisely g pairs of curves suffices to define the attaching homeomorphism of two genus- g surfaces. Then we demonstrate certain Heegaard splittings of man­ifolds such as S 3, S 2 x S 1 , and T 3 and offer techniques to visualize them. We observe a simple classification of compact, closed, orientable 3-manifolds by Heegaard genus. Manifolds that admit genus-1 splittings are also lens spaces, which can be defined as particular quotient spaces of the 3-sphere S 3 . Finally we introduce Dehn surgery, a method by which any compact, closed, orientable 3-manifold can be obtained. Dehn surgery on S 3 along some knot or link K entails removing an open tubular neighborhood N ( K ), defining a homeomor-phism of δ ¯ N ( K ), and attaching the new neighborhood ¯ N ′ ( K ) to the boundary of the complement of N ( K ) in S 3 . In general, Dehn surgery along the un-knot produces a lens space. Thus we discuss methods of obtaining, visualizing, and classifying 3-manifolds and the connections between surgery and Heegaard splittings. Topology Heegaard splitting 3-manifolds surgery lens space Dehn surgery Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7435624","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Systematic Review","associatedPublications":[],"authors":[{"id":504291564,"identity":"a9632880-76ca-4c36-a7e8-21b4a0757316","order_by":0,"name":"Christopher de la Viesca","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA7UlEQVRIiWNgGAWjYFCCBMYDDAwSDPwMzA0MIAZQhIGBB78WBrAWyQZG0rQwMBgcAGthIKyFvz35wIGfbRZyxscPtknd3GHBYM6ewPjgbRtuLRJnniUc7G2TMDY7k9gmnXtGgsGy5wGz4Vw8WgwkcgwO8JyRSNx2AKSlTYLB4EYCmzQvXi35Hw7+OSNRv7n/IVwL+2/8WnIYDvNUSCQYSCDZwoxPC9AvBodlKiQMZ9x42GwN1MJjcOZhs+Scc7i1AEPs4cM3BnXy/P3JB2/nttXJGRxPPvjhTRluLRgAGCPwCBoFo2AUjIJRQC4AAJkSUlEED9cjAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0009-0000-1779-8918","institution":"University of Notre Dame","correspondingAuthor":true,"prefix":"","firstName":"Christopher","middleName":"de la","lastName":"Viesca","suffix":""}],"badges":[],"createdAt":"2025-08-22 15:05:48","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-7435624/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7435624/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":89786153,"identity":"8e6c8a1c-b7c2-4ea4-8690-86eb498615e6","added_by":"auto","created_at":"2025-08-25 04:15:08","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":792646,"visible":true,"origin":"","legend":"","description":"","filename":"HeegaardSplittingsandSurgeryon2and3Manifolds4.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7435624/v1_covered_8afef07c-b1ce-4a10-8089-fea5d62ae8f5.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eHeegaard Splittings and Surgery on 2- and 3-Manifolds\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Heegaard splitting, 3-manifolds, surgery, lens space, Dehn surgery","lastPublishedDoi":"10.21203/rs.3.rs-7435624/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7435624/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eA Heegaard splitting of a 3-manifold is a representation of it as the union of two handlebodies with the same boundary. Each splitting is defined by an attaching homeomorphism between the boundaries of these two handle-bodies. We discuss surgery on surfaces (2-manifolds) to explain why specifying precisely \u003cem\u003eg \u003c/em\u003epairs of curves suffices to define the attaching homeomorphism of two genus-\u003cem\u003eg \u003c/em\u003esurfaces. Then we demonstrate certain Heegaard splittings of man­ifolds such as \u003cem\u003eS\u003c/em\u003e3, \u003cem\u003eS\u003c/em\u003e\u003csup\u003e2\u003c/sup\u003e x \u003cem\u003eS\u003c/em\u003e\u003csup\u003e1\u003c/sup\u003e, and \u003cem\u003eT\u003c/em\u003e\u003csup\u003e3\u003c/sup\u003e and offer techniques to visualize them. We observe a simple classification of compact, closed, orientable 3-manifolds by Heegaard genus. Manifolds that admit genus-1 splittings are also lens spaces, which can be defined as particular quotient spaces of the 3-sphere \u003cem\u003eS\u003c/em\u003e\u003csup\u003e3\u003c/sup\u003e. Finally we introduce Dehn surgery, a method by which any compact, closed, orientable 3-manifold can be obtained. Dehn surgery on \u003cem\u003eS\u003c/em\u003e\u003csup\u003e3\u003c/sup\u003e along some knot or link \u003cem\u003eK \u003c/em\u003eentails removing an open tubular neighborhood \u003cem\u003eN\u003c/em\u003e(\u003cem\u003eK\u003c/em\u003e), defining a homeomor-phism of \u003cem\u003eδ \u003c/em\u003e\u003csup\u003e¯\u003c/sup\u003e\u003cem\u003eN\u003c/em\u003e(\u003cem\u003eK\u003c/em\u003e), and attaching the new neighborhood \u003csup\u003e¯\u003c/sup\u003e\u003cem\u003eN\u003c/em\u003e\u003csup\u003e\u003cem\u003e′\u003c/em\u003e\u003c/sup\u003e(\u003cem\u003eK\u003c/em\u003e) to the boundary of the complement of \u003cem\u003eN\u003c/em\u003e(\u003cem\u003eK\u003c/em\u003e) in \u003cem\u003eS\u003c/em\u003e\u003csup\u003e3\u003c/sup\u003e. In general, Dehn surgery along the un-knot produces a lens space. Thus we discuss methods of obtaining, visualizing, and classifying 3-manifolds and the connections between surgery and Heegaard splittings.\u003c/p\u003e","manuscriptTitle":"Heegaard Splittings and Surgery on 2- and 3-Manifolds","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-25 04:06:59","doi":"10.21203/rs.3.rs-7435624/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"e0ef8f88-b331-4a88-bc0d-2799dbbcd7e1","owner":[],"postedDate":"August 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":53635841,"name":"Topology"}],"tags":[],"updatedAt":"2025-08-25T04:07:00+00:00","versionOfRecord":[],"versionCreatedAt":"2025-08-25 04:06:59","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-7435624","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7435624","identity":"rs-7435624","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}