Seiberg-Witten Invariants, Alexander Polynomials, and Fibred Classes

TL;DR

This paper explores how Seiberg-Witten invariants can determine when 3-manifolds fiber over a circle and constructs examples of surface bundles with spin 4-manifold total spaces.

Contribution

It provides new criteria using Seiberg-Witten invariants for fibering of 3-manifolds and constructs explicit examples of surface bundles with specific properties.

Findings

Identified conditions for 3-manifolds to fiber over a circle

Constructed genus 1 and 2 surface bundles as spin 4-manifolds

Demonstrated applications of Seiberg-Witten invariants in 4-manifold topology

Abstract

Since their introduction in 1994, the Seiberg-Witten invariants have become one of the main tools used in 4-manifold theory. In this thesis, we will use these invariants to identify sufficient conditions for a 3-manifold to fibre over a circle. Additionally, we will construct several examples of genus 1 and 2 surface bundles and prove their total spaces are spin 4-manifolds.