Georgia Institute of Technology College of Sciences

4-manifold topology is characterized by unexpected differences between the smooth and topological categories. For instance, it is the only dimension where there can exist infinitely many manifolds $Y_i$ which are homeomorphic to but not diffeomorphic to $X$. A natural question: how does one construct examples of this phenomenon? In this talk, we focus on the method of reverse engineering, which allows for the construction of “small” exotic 4-manifolds. Surprisingly, symplectic geometry is the main ingredient that makes this approach work! We survey the known results related to reverse engineering, and try to pinpoint an error in a paper of Akhmedov-Park, which claimed the existence of an exotic $S^2 \times S^2$.