The 191 Orientable Octahedral Manifolds

We enumerate all spaces obtained by gluing in pairs the faces of the octahedron in an orientation-reversing fashion. Whenever such a gluing gives rise to non-manifold points, we remove small open neighbourhoods of these points, so we actually deal with three-dimensional manifolds with (possibly empty) boundary. There are 298 combinatorially inequivalent gluing patterns, and we show that they define 191 distinct manifolds, of which 132 are hyperbolic and 59 are not. All the 132 hyperbolic manifolds were already considered in different contexts by other authors, and we provide here their known “names” together with their main invariants. We also give the connected sum and JSJ decompositions for the 59 non-hyperbolic examples. Our arguments make use of tools coming from hyperbolic geometry, together with quantum invariants and more classical techniques based on essential surfaces. Many (but not all) proofs were carried out by computer, but they do not involve issues of numerical accuracy. MSC (2000): 57M50 (primary), 57M25 (secondary). At the very beginning of his fundamental book [21], as an example of the richness of topology in three dimensions, Bill Thurston mentioned the fact that there are quite a few inequivalent ways of gluing together in pairs the faces of the octahedron. However, to our knowledge, as of today nobody had ever exactly determined the number of non-homeomorphic 3-manifolds arising as the results of these gluings. In this note we give a full solution to this problem, in the context of orientable (but unoriented) manifolds. After proving that there are 298 inequivalent gluing patterns, we have in fact proved the following: