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A Two Dimensional Description of Heegaard Splittings

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dc.contributor.advisor Sullivan, Dennis en_US
dc.contributor.author Sadanand, Chandrika en_US
dc.contributor.other Department of Mathematics en_US
dc.date.accessioned 2017-09-20T16:50:09Z
dc.date.available 2017-09-20T16:50:09Z
dc.date.issued 2017-05-01 en_US
dc.identifier.uri http://hdl.handle.net/11401/76383 en_US
dc.description 69 pgs en_US
dc.description.abstract 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 Poincaré 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 sewn together to make a statement for all Heegaard splittings and arbitrary compact orientable 3-manifolds. en_US
dc.description.sponsorship This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree. en_US
dc.format Monograph en_US
dc.format.medium Electronic Resource en_US
dc.language.iso en_US en_US
dc.publisher The Graduate School, Stony Brook University: Stony Brook, NY. en_US
dc.subject.lcsh Mathematics en_US
dc.subject.other 3-manifold, Heegaard en_US
dc.title A Two Dimensional Description of Heegaard Splittings en_US
dc.type Dissertation en_US
dc.mimetype Application/PDF en_US
dc.contributor.committeemember Chas, Moira en_US
dc.contributor.committeemember Plamenevskaya, Olga en_US
dc.contributor.committeemember RoÄ ek, Martin en_US

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