Analytical results on Seiberg-Witten equations on homology S¹ × S³

Abstract

This thesis studies the Seiberg-Witten equation on a periodic 4-manifold obtained from a homology S¹ × S³. The first chapter is a brief introduction of the topic. The second chapter reviews basic preliminaries on Seiberg-Witten equations. The third chapter proves the compactness given a uniform L^2-bound on curvature and a C^0-bound on the spinor. The fourth chapter reconstructs the Bauer-Furuta invariant with analysis, showing that the finite-dimensional approximation map will recover the full configuration space and the Seiberg-Witten map. The fifth chapter generalizes the fourth chapter into the periodic manifold setting, showing that the extra boundary conditions will impose an exponentially decaying factor in the higher modes.