3-dimensional Seiberg-Witten invariants and non-Kählerian geometry

Abstract

The paper is concerned with Seiberg-Witten invariants of closed 3-manifolds and their relation to the 4-dimensional invariants. The main points are: 1. The introduction of parameter-dependent Seiberg-Witten invariants for 3-manifolds with b1 = 0. These invariants depend on a complicated chamber structure defined by a differential geometric Brill-Noether locus in the parameter space. 2. A comparison theorem which shows that the Seiberg-Witten invariants of 3-manifolds M with b_1(M ) ≥ 1 are simply the restrictions of the 4-dimensional invariants of the product S^1 × M to the set of pull-back Spinc-structures. As an immediate consequence one gets the wall crossing formula for the invariants of 3-dimensional manifolds with b1 = 1 from the 4-dimensional wall crossing formula for manifolds with b+ = 1. 3. A novel method to compute the Seiberg-Witten invariants of 3-manifolds M using (non-Kahlerian) complex geometry on S 1 × M , provided this product has such a structure which is compatible with a product metric and orientation induced from M . This method relies essentially on a Kobayashi-Hitchin correspondence, which is new in the non-Kahler case. 4. A complete description of those 3-manifolds to which our method applies. These are Riemannian manifolds which admit an oriented Riemannian foliation by geodesics. Such manifolds are always diffeomorphic to Seifert manifolds, but the foliation has not to be Seifert; it can have non-closed leaves. 5. Explicit computations of Seiberg-Witten invariants for the 3-sphere S^3 using non-Seifert Riemannian foliations. We show that the chamber structure induced by the Brill-Noether locus is very complicated. There exist countably many dif- ferent chambers, and the Seiberg-Witten invariant takes on all integer values.