Algebra/Topology seminar Arunima Ray

Abstract: Unlike in higher ambient dimensions, maps of surfaces to 4-manifolds are not generically embeddings. The minimum genus of a smoothly embedded surface representing a given second homology class in a simply connected 4-manifold can be arbitrarily high. I will show that in contrast every primitive second homology class in a closed simply connected 4-manifold is represented by a locally flat embedded torus. This is simultaneously a special case of an old result of Lee-Wilczynski and a new result joint with Daniel Kasprowski, Mark Powell, and Peter Teichner. No prior knowledge of 4-manifolds will be needed and most arguments will be direct and visual.