Several Compactness Results in Gauge Theory and Low Dimensional Topology

Abstract

This thesis studies several compactness problems in gauge theory and explores their applications is low dimensional topology. The first chapter studies a connection between taut foliations and Seiberg-Witten theory. Let Y be a closed oriented 3-manifold and F a smooth oriented foliation on Y. Assume that F does not admit any transverse invariant measure. This chapter constructs an invariant c(F) for F which takes value in the monopole Floer homology group of Y. The invariant is well defined up to a sign. The second chapter proves the deformation invariance for the parity of the number of Klein-bottle leaves in a smooth taut foliation. Given two smooth cooriented taut foliations, assume that every Klein-bottle leaf has non-trivial linear holonomy, and assume that the two foliations can be smoothly deformed to each other through taut foliations, then the parities of the number of Klein-bottle leaves are the same. The third chapter proves that the zero locus of a Z/2 harmonic spinor on a 4 dimensional manifold is 2-rectifiable and has locally finite Minkowski content.