Exotic 4-manifolds homotopy equivalent to S2

In this thesis, we will be reviewing morse theory and handle decompositions for manifolds while paying special attention to 4-manifolds and related notions such as handles moves and Kirby Calculus. We will also review the fundamentals of the relationship of knot theory and dimension four. The main objective will be to present an exotica result: There are exotic 4-manifolds with boundary that are homotopy equivalent to S2, this result is due to Akbulut. I will not be presenting the original proof. Instead, we will be mainly following a different approach. In particular, I will be also following the lectures delivered by Piccirilo at Tech Topology Summer School and the lectures delivered at Singularities and Low-Dimensional Topology Winter School, Alfred Renyi Institute of Mathematics.