Incompressible surfaces and Heegaard splittings of 3-manifolds

Abigail Thompson (Davis and IAS)

Abstract: Let M be a closed orientable 3-manifold. A Heegaard surface F splits M into two handlebodies. If M is a reducible 3-manifold, Haken showed that one can find a reducing sphere intersecting F in a single simple closed curve. What if M is irreducible but toroidal In general the intersection of an essential torus with F can be quite complicated, so it can be difficult to determine from a Heegaard splitting whether or not M contains an essential torus. We provide some conditions under which this determination can be made.