Type
Text
Type
Dissertation
Advisor
Sullivan, Dennis | Chas, Moira | Plamenevskaya, Olga | RoÄ ek, Martin.
Date
2017-05-01
Keywords
Mathematics | 3-manifold, Heegaard
Department
Department of Mathematics
Language
en_US
Source
This work is sponsored by the Stony Brook University Graduate School in compliance with the requirements for completion of degree.
Identifier
http://hdl.handle.net/11401/76383
Publisher
The Graduate School, Stony Brook University: Stony Brook, NY.
Format
application/pdf
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. | 69 pages
Recommended Citation
Sadanand, Chandrika, "A Two Dimensional Description of Heegaard Splittings" (2017). Stony Brook Theses and Dissertations Collection, 2006-2020 (closed to submissions). 2307.
https://commons.library.stonybrook.edu/stony-brook-theses-and-dissertations-collection/2307