Triangulated Manifolds with Few Vertices: Geometric 3-Manifolds

The understanding and classication of (compact) 3-dimensional manifolds (without boundary) is with no doubt one of the most prominent tasks in topology ever since Poincare's fundamental work [88] on l'analysis situs appeared in 1904. There are various ways for constructing 3-manifolds, some of which that are general enough to yield all 3-manifolds (orientable or nonorientable) and some that produce only particular types or classes of examples. According to Moise [73], all 3-manifolds can be triangulated. This im-plies that there are only countably many distinct combinatorial (and therefore at most so many dierent topological) types that result from gluing together tetrahedra. Another way to obtain 3-manifolds is by starting with a solid 3-dimensional polyhedron for which surface faces are pairwise identied (see, e.g., Seifert [98] and Weber and Seifert [118]). Both approaches are rather general and, on the rst sight, do not give much control on the kind of manifold we can expect as an outcome. However, if we want to determine the topological typ