Cerf Theory and Trisections of Four Manifolds

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Dadsetan, Ali | 2019

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Type of Document: M.Sc. Thesis
Language: Farsi
Document No: 53317 (02)
University: Sharif University of Technology
Department: Mathematical Sciences
Advisor(s): Bahraini, Alireza
Abstract:
Morse 2-functions are higher-dimensional analogs to morse functions. A morse 2-function is a mapping to 2-dimensional disk with its critical points satisfying certain generality conditions. The goal of defining morse 2-functions and trisections is to generalize methods of 3-dimensional manifolds to dimension 4. A morse function on a 3-dimensional manifold leads to a decomposition of that manifold to two manifolds with boundary with common boundary. This decomposition is called Heegaard splitting. This decomposition is unique up to stabilization. Trisections are 4-dimensional analogs of Heegaard splittings and decompose manifold into three parts. The intersection of each pair is a solid genus n tori, and the intersection of all three components is a genus g surface. Each Heegaard splitting comes from a morse function and a natural partitioning of the unit interval. Similarly, each trisection comes from a morse 2-function and partitioning of the unit disk into three chunks. The existence and uniqueness theorems about these objects lead to a framework of defining invariants for four-dimensional Manifolds
Keywords:
Singularity ; Four Dimensional Manifolds ; Invariant Manifolds ; Morse 2-Functions

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