Georgia Institute of Technology College of Sciences

We will show that every closed orientable 3-manifold bounds an orientable 4-manifold. If time permits, we will also see an application to embedding closed orientable 3-manifolds to R^5.

Given a surface, intersection information about the simple closed curves on the surface is encoded in its curve graph. Vertices are homotopy classes of curves, and edges connect vertices corresponding to curves with disjoint representatives. We can wonder what subgraphs of the curve graph are possible for a given surface. For example, if we fix a surface, then a graph with sufficiently large clique number cannot be a subgraph of its curve graph. This is because there are only so many distinct and mutually disjoint curves in a given surface. We will discuss a new obstruction to a graph being a subgraph of individual curve graphs given recently by Bering, Conant, and Gaster.

In 1988, Penner conjectured that all pseudo-Anosov mapping classes arise up to finite power from a construction named after him. This conjecture was known to be true on some simple surfaces, including the torus, but has otherwise remained open. I will sketch the proof (joint work with Hyunshik Shin) that the conjecture is false for most surfaces.

We will describe some recent work of Lidman, Sivek, and Yasui as it pertains to the cabling conjecture. This is a question about which Dehn surgeries in S^3 yeild reducible 3-manifolds.