The Seiberg-Witten theory of homology 3-spheres

Abstract

In this thesis we study the Seiberg-Witten theory of an oriented homology 3-sphere. The goal is to extract topological invariants - the Seiberg-Witten invariants - by counting the solutions to the Seiberg-Witten equations on the manifold. The first question we consider is whether the Seiberg-Witten invariants depend on the geometric or analytic data involved in their definition. In the first main result of this thesis, we completely determine the dependence of the Seiberg-Witten invariants on the data involved in their definition. In particular, we show that even for the simplest manifold, the 3-sphere $S^3$, the Seiberg-Witten invariants take infinitely many different values. The rest of this thesis is devoted to understanding the Seiberg-Witten invariants in a specific geometric setting - the surgery setting. In that context we prove a gluing formula, which identifies the Seiberg-Witten invariants as certain ``homological intersection numbers''.

An incorrect statement (the surgery formula) in an earlier version is removed. This version is the PhD thesis of the author

Cited by in corpus (3)

• Associative submanifolds of a G2 manifold
• Calibrated Manifolds and Gauge Theory
• On the Relationship between the Rozansky-Witten and the 3-Dimensional Seiberg-Witten Invariants