The Seiberg-Witten theory of homology 3-spheres

Details The Seiberg-Witten theory of homology 3-spheres The Seiberg-Witten theory of homology 3-spheres

1997-03-16

arxiv.org/abs/dg-ga/9703009v4

Mathematics

Differential Geometry

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

Scientific paper

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''.

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