CEU Electronic Theses and Dissertations, 2023
| Author | Mihajlović, Stefan |
|---|---|
| Title | Elliptic Fibrations in 4-Dimensional Topology and Geometry |
| Summary | This thesis deals with various phenomena arising in smooth 4-dimensional topology. The connecting thread between 3 parts are elliptic fibrations - a structure which helps us represent some 4-manifolds as 2 dimensional families of tori, with some of the tori degenerating into interesting singular surfaces. Part I of the thesis emphasises the difference between smooth and topological 4- dimensional worlds by constructing exotic smooth structures on the same underlying topological space. The construction is inspired by a similar construction which uses elliptic fibrations. Part II is concerned with analyzing interesting 4-dimensional spaces which have very rich geometry, and come equipped with an elliptic fibration. These are moduli spaces of certain meromorphic Higgs bundles with an underlying Hitchin’s fibration and we analyse a class of cases when this fibration has a special fiber called an E6-tilda fiber. Part III deals with an interesting application of elliptic fibrations different from the first part. We use configurations of spheres to understand which knots in the boundary of a small 4-ball in a 4-manifold, bound a disk in the interior of the manifold. Even though we prove a general result, more specifically, using a configuration of 22 spheres, we show that many complicated knots bound disks in the 𝐾3-surface. To prove there are 22 spheres in the 𝐾3-surface, incidentaly, we can use an elliptic fibration with 3 ̃E6-tilda fibers, but the analogue for elliptic surfaces &#x 1d438;(𝑛 ;) requires a more complicated combination of singular fibers. |
| Supervisor | Stipsicz, András; Szabó, Szilárd |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2023/mihajlovic_stefan.pdf |
Visit the CEU Library.
© 2007-2021, Central European University