|Abstract: Many fundamental topological problems about 3-manifolds are algorithmically solvable in theory but continue to withstand practical computations. In recent years some of these problems have been shown to allow efficient solutions, if the input 3-manifold comes with a sufficiently "thin" presentation.|
More specifically, a 3-manifold given as a triangulation is considered thin, if the treewidth of its dual graph is small. I will show how this combinatorial parameter, defined on a triangulation, can be linked back to purely topological properties of the underlying manifold. From this connection it can then be followed that, for some 3-manifolds, we cannot hope for a thin triangulation.