Mathematics Colloquia and Seminars

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In this talk, we will take a look at Haken manifolds, which form a large and quite well-understood class of 3-manifolds. Loosely put, a Haken manifold is a 3-manifold which contains an embedded (essential) surface that is characteristic for its topology. After discussing how one determines if a given 3-manifold is a Haken manifold, we will spend the remainder of the time looking at hierarchies, which are among the most important tools used to study Haken manifolds. Hierarchies arise from the idea that one can often obtain information about a space by cutting it into successively simpler pieces, and they provide an often-convenient inductive method of proof. After overviewing a couple of major results that have been obtained through the use of hierarchies, we will sketch Waldhausen's proof of the fact that every Haken manifold has a hierarchy. If time allows, we will briefly mention some further results concerning how one can control the length of a hierarchy.