Citation
Jeong, Gahye (2018) SelfGluing Formula of the Monopole Invariant and its Application on Symplectic Structures. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BH06KS91. https://resolver.caltech.edu/CaltechTHESIS:05252018080955604
Abstract
SeibergWitten theory has been an important tool in studying a class of 4manifolds. Moreover, the SeibergWitten invariants have been used to compute for simple structures of symplectic manifolds. The normal connected sum operation on 4 manifolds has been used to construct 4manifolds. In this thesis, we demonstrate how to compute the SeibergWitten invariant of 4manifolds obtained from the normal connected sum operation. In addition, we introduce the application of the formula on the existence of symplectic structures of manifolds given by the normal connected sum.
In Chapter 1, we study the SeibergWitten theory for various types of 3 and 4 manifolds. We review the SeibergWitten equation and invariants for 4manifolds with cylindrical ends as well as closed and smooth 4manifolds . Furthermore, we explain how to compute the SeibergWitten invariants for two types of 4manifolds: the products of a circle and a 3manifold and sympectic manifolds.
In Chapter 2, we prove that the SeibergWitten invariant of a new manifold obtained from the normal connected sum can be represented by the SeibergWitten invariant of the original manifolds. In [Tau01], the author has proved the case of the operation along tori. In [MST96], the authors have proved the case of the operation along surfaces with genus at least 2 when the product of the circle and the surface is separating in the ambient 4manifold. In this thesis, we show the proof of the remaining case.
In Chapter 3, we prove the existence of certain symplectic structures on manifolds obtained from the normal connected sum of two 4manifolds using the multiple gluing formula stated in Chapter 2. We explain how to construct covering spaces of the manifold and compute the SeibergWitten invariant of the covering spaces by the gluing formula. From the relation between the SeibergWitten invariants and symplectic structures, we prove the main application.
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  4manifold, monopole invariant, geometry, symplectic structure  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Minor Option:  Computational Science and Engineering  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  26 April 2018  
NonCaltech Author Email:  ghjeong0717 (AT) gmail.com  
Record Number:  CaltechTHESIS:05252018080955604  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:05252018080955604  
DOI:  10.7907/BH06KS91  
ORCID: 
 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  10934  
Collection:  CaltechTHESIS  
Deposited By:  Gahye Jeong  
Deposited On:  25 May 2018 19:00  
Last Modified:  05 Jul 2022 19:06 
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