{"@context":"http://iiif.io/api/presentation/2/context.json","@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/manifest.json","@type":"sc:Manifest","label":"A Two Dimensional Description of Heegaard Splittings","metadata":[{"label":"dc.description.sponsorship","value":"This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree."},{"label":"dc.format","value":"Monograph"},{"label":"dc.format.medium","value":"Electronic Resource"},{"label":"dc.identifier.uri","value":"http://hdl.handle.net/11401/76383"},{"label":"dc.language.iso","value":"en_US"},{"label":"dc.publisher","value":"The Graduate School, Stony Brook University: Stony Brook, NY."},{"label":"dcterms.abstract","value":"Consider the following two ways to decompose 3-manifolds: (i) A compact 3-manifold can be decomposed by a Heegaard splitting into two well-understood, homeomorphic manifolds, glued along their boundary. (ii) A compact orientable 3-manifold can be decomposed uniquely as a connect sum of prime 3-manifolds. Stallings described Heegaard splittings using classes of continuous maps between surfaces and two dimensional complexes. He studied the Poincar\u00c3\u00a9 conjecture with these maps using group theory. This dissertation considers these maps more literally, using geometric and topological arguments. One might wonder if these maps fold the surface, or crush handles. We find that minimal genus Heegaard splittings of prime 3-manifolds (with a couple exceptions) can be described by locally injective two dimensional maps. These locally injective maps induce families of conformal structures, and also square complex structures, on the domain surface. The 3-manifold, together with a minimal Heegaard splitting, can be recovered from any member of a family. The construction on primes can be \u00e2\u20ac\u0153sewn\u00e2\u20ac  together to make a statement for all Heegaard splittings and arbitrary compact orientable 3-manifolds."},{"label":"dcterms.available","value":"2017-09-20T16:50:09Z"},{"label":"dcterms.contributor","value":"Plamenevskaya, Olga"},{"label":"dcterms.creator","value":"Sadanand, Chandrika"},{"label":"dcterms.dateAccepted","value":"2017-09-20T16:50:09Z"},{"label":"dcterms.dateSubmitted","value":"2017-09-20T16:50:09Z"},{"label":"dcterms.description","value":"Department of Mathematics"},{"label":"dcterms.extent","value":"69 pg."},{"label":"dcterms.format","value":"Monograph"},{"label":"dcterms.identifier","value":"http://hdl.handle.net/11401/76383"},{"label":"dcterms.issued","value":"2017-05-01"},{"label":"dcterms.language","value":"en_US"},{"label":"dcterms.provenance","value":"Made available in DSpace on 2017-09-20T16:50:09Z (GMT). No. of bitstreams: 1\nSadanand_grad.sunysb_0771E_13311.pdf: 1085779 bytes, checksum: 1326fe935b59a4ef35e1c3fa371cd36a (MD5)\n  Previous issue date:    1"},{"label":"dcterms.publisher","value":"The Graduate School, Stony Brook University: Stony Brook, NY."},{"label":"dcterms.subject","value":"3-manifold, Heegaard"},{"label":"dcterms.title","value":"A Two Dimensional Description of Heegaard Splittings"},{"label":"dcterms.type","value":"Dissertation"},{"label":"dc.type","value":"Dissertation"}],"description":"This manifest was generated dynamically","viewingDirection":"left-to-right","sequences":[{"@type":"sc:Sequence","canvases":[{"@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/canvas/page-1.json","@type":"sc:Canvas","label":"Page 1","height":1650,"width":1275,"images":[{"@type":"oa:Annotation","motivation":"sc:painting","resource":{"@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/66%2F26%2F01%2F66260154630208321344523581121338805489/full/full/0/default.jpg","@type":"dctypes:Image","format":"image/jpeg","height":1650,"width":1275,"service":{"@context":"http://iiif.io/api/image/2/context.json","@id":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/66%2F26%2F01%2F66260154630208321344523581121338805489","profile":"http://iiif.io/api/image/2/level2.json"}},"on":"https://repo.library.stonybrook.edu/cantaloupe/iiif/2/canvas/page-1.json"}]}]}]}