| A well-known conjecture in low-dimensional topology asserts that if M is a closed, oriented 3-manifold such that the 4-dimensional manifold S 1 x M admits a symplectic form, namely, a closed 2-form o such that o ∧ o is nowhere zero, then M fibers over the circle. This dissertation presents a geometric approach to proving this conjecture using Seiberg-Witten gauge theory. To elaborate, our approach entails the study of a one-parameter family of partial differential equations on M parametrized by S1. These equations are obtained by perturbing the 3-dimensional Seiberg-Witten equations on M via a closed 2-form that comes from the symplectic form o on S1 x M. Under favorable conditions, a general philosophy due to Taubes lets us construct a one-parameter family of 1-dimensional submanifolds of M using solutions of the previously mentioned family of equations. Moreover, these 1-dimensional submanifolds of M give a current in S1 x M with unique properties. We use the existence of such a current to derive a fundamental contradiction assuming that the conjecture is false. In particular, we give a proof of the conjecture when the first Betti number of M is equal to 1 and (S1 x M, o) has non-torsion anticanonical class. |
