The Theta Divisor and Three-Manifold Invariants

TL;DR

This paper introduces a new invariant for oriented three-manifolds with positive first Betti number, based on Heegaard splittings and the theta divisor, linking it to classical torsion invariants and Seiberg-Witten theory.

Contribution

It defines a novel three-manifold invariant using complex geometric tools and relates it to existing torsion and Seiberg-Witten invariants.

Findings

The invariant is well-defined for manifolds with b_1 > 0.

It is shown to coincide with classical torsion invariants.

The relationship with Seiberg-Witten theory is established.

Abstract

In this paper we study an invariant for oriented three-manifolds with $b_1>0$, which is defined using Heegaard splittings and the theta divisor of a Riemann surface. The paper is divided into two parts, the first of which gives the definition of the invariant, and the second of which identifies it with more classical (torsion) invariants of three-manifolds. Its close relationship with Seiberg-Witten theory is also addressed.