Topological index theory for surfaces in 3-manifolds

We utilize the curve complex to define an isotopy invariant index of a surface in a 3-manifold, and show that it mimics the index of a (unstable) minimal surface. This allows us to find purely topological analogues to familiar techniques from geometry, such as barrier arguments. Deep results follow concerning Heegaard splittings, bridge positions of knots, and normal surfaces. In addition, this new viewpoint opens up a host of exciting new questions for the field of 3-manifold topology.