Self-Gluing Formula of the Monopole Invariant and its Application on Symplectic Structures

Citation

Jeong, Gahye (2018) Self-Gluing Formula of the Monopole Invariant and its Application on Symplectic Structures. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BH06-KS91. https://resolver.caltech.edu/CaltechTHESIS:05252018-080955604

Abstract

Seiberg-Witten theory has been an important tool in studying a class of 4-manifolds. Moreover, the Seiberg-Witten 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 4-manifolds. In this thesis, we demonstrate how to compute the Seiberg-Witten invariant of 4-manifolds 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 Seiberg-Witten theory for various types of 3- and 4- manifolds. We review the Seiberg-Witten equation and invariants for 4-manifolds with cylindrical ends as well as closed and smooth 4-manifolds . Furthermore, we explain how to compute the Seiberg-Witten invariants for two types of 4-manifolds: the products of a circle and a 3-manifold and sympectic manifolds.

In Chapter 2, we prove that the Seiberg-Witten invariant of a new manifold obtained from the normal connected sum can be represented by the Seiberg-Witten 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 4-manifold. 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 4-manifolds using the multiple gluing formula stated in Chapter 2. We explain how to construct covering spaces of the manifold and compute the Seiberg-Witten invariant of the covering spaces by the gluing formula. From the relation between the Seiberg-Witten invariants and symplectic structures, we prove the main application.

Thesis Files