Exotic constructions on covers branched over hyperplane arrangements

Abstract

The purpose of this thesis is to study and construct geometric structures on 4-manifolds from a starting point of combinatorial objects.

In part as a consequence of embedded surfaces and codimension two submanifolds coinciding in dimension four, many of the tools that are used in higher dimensions fail or are underwhelming when applied to 4-manifolds. For this reason, the development and advancement of techniques that are applicable to 4-manifolds are of particular interest and importance to low dimensional topologists. The general techniques of interest are those that either construct a 4-manifold in a novel way or find interesting embedded submanifolds.

Consequently, we shall contribute to the study of 4-manifolds by investigating ways to construct 4-manifolds with positive signature in addition to describing construction methods that can guarantee the existence of algebraically interesting embedded symplectic submanifolds.

In this thesis we investigate how the combinatorial data of line arrangements and the algebraic data of their complements in rational surfaces can be utilized to construct symplectic 4-manifolds with arbitrarily large signatures through the method of branched coverings. In general, we not only show that these line arrangements can be used to provide asymptotic bounds for the existence of symplectic 4-manifolds but we also show that for any line arrangement, there exist symplectic branched covers with sufficiently nice geometric and topological properties. Namely, we show they contain embedded symplectic surfaces which carry their fundamental group.