Floer Homology and Symplectic Forms on S1 s M3.

[摘要] A well-known conjecture in low-dimensional topology asserts that if $M$ is a closed, oriented 3-manifold such that the 4-dimensional manifold $S^1 times M$ admits a symplectic form, namely, a closed 2-form $omega$ such that $omega wedge omega$ 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 $S^1$. 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 $omega$ on $S^1 times M$. 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 $S^1 times M$ 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 $(S^1 times M,omega)$ has non-torsion anticanonical class.

[发布机构] University of Michigan

[效力级别] Symplectic Form

[关键词] Floer Homology;Symplectic Form;Mathematics;Science;Mathematics