Abstract:

Essential surfaces in 3-manifolds play a fundamental role in the study of 3-manifold topology. Cutting a 3-manifold along essential surfaces allows us to simplify the 3-manifold and gain a better insight into its global topological structure. The annulus/torus decomposition due to Johannson, Jaco and Shalen and Thurston's hyperbolization theorem together give a unique way to decompose a 3-manifold into three fundamental pieces: I-bundles, Seifert 3-manifolds, and hyperbolic 3-manifolds. In this talk, we will have a quick tour through these two celebrated theorems, and some applications to the theory of handlebody-knots.