Constructing locally flat surfaces in 4-manifolds (part 1)

There are two main approaches to building locally flat surfaces in 4-manifolds: direct methods applying Freedman-Quinn's disc embedding theorem, and indirect methods using surgery theory. (Notably the second method also requires the disc embedding theorem, but only indirectly.) In this sequence of two lectures, I will give an introduction to both methods. In this first lecture I will give a direct, constructive proof of a result of Lee-Wilczynski which states that every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat torus.