ETD

• 4-manifolds
• aspherical manifolds
• exotic manifolds

The study of exotic structures is a central theme in 4-dimensional topology, with examples obtained through a wide range of techniques. However, these constructions are often ill-suited to closed aspherical manifolds, as topological surgery typically creates nontrivial spheres.
In this thesis, we provide an explicit example of a pair of closed, aspherical 4-manifolds that are homeomorphic but not diffeomorphic, following a paper by M. Davis, K. Hayden, J. Huang, D. Ruberman and N. Sunukjian. We begin by reviewing some fundamental topological tools, such as handle decompositions and the representation of 4-manifolds via Kirby diagrams. Using these tools, we construct a pair of compact, aspherical manifolds with boundary starting from the Akbulut cork and its exotic copy. We then describe the reflection group trick, a technique developed by M. Davis that utilizes right-angled Coxeter groups to turn a compact, aspherical manifold with boundary into a closed, aspherical manifold. Finally, smooth structures are distinguished by gauge-theoretical arguments, specifically the adjunction inequality, to give bounds on the genus of smoothly embedded surfaces.