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We compute for all orientable irreducible geometric 3-manifolds certain complexity functions that approximate from above Matveev's natural complexity, known to be equal to the minimal number of tetrahedra in a triangulation. We can show that the upper bounds on Matveev's complexity implied by our computations are sharp for thousands of manifolds, and we conjecture they are for infinitely many, including all Seifert manifolds. Our computations and estimates apply to all the Dehn fillings of M613(the complement of the three-component chain-link, conjectured to be the smallest triply cusped hyperbolic manifold), whence to infinitely many among the smallest closed hyperbolic manifolds. Our computations are based on the machinery of the decomposition into ‘bricks’ of irreducible manifolds, developed in a previous paper. As an application of our results we completely describe the geometry of all 3-manifolds of complexity up to 9.

We classify the orientable finite-volume hyperbolic 3-manifolds having nonempty compact totally geodesic boundary and admitting an ideal triangulation with at most four tetrahedra. We also compute the volume of all such manifolds, describe their canonical Kojima decomposition, and discuss manifolds having cusps.The eight manifolds built from one or two tetrahedra were previously known. There are 151 different manifolds built from three tetrahedra, realizing 18 different volumes. Their Kojima decomposition always consists of tetrahedra (but occasionally requires four of them). There is a single cusped manifold, which we can show to be a knot complement in a genus-2 handlebody. Concerning manifolds built from four tetrahedra, we show that there are 5,033 different ones, with 262 different volumes. The Kojima decomposition consists either of tetrahedra (as many as eight of them in some cases), of two pyramids, or of a single octahedron. There are 30 manifolds having a single cusp and one having two cusps.Our results were obtained with the aid of a computer. The complete list of manifolds (in SnapPea format) and full details on their invariants are available on the world wide web.

We describe an algorithm which has enabled us to give a complete list, without repetitions, of all closed oriented irreducible three-manifolds of complexity up to 9. More interestingly, we have actually been able to give a name to each such manifold, by recognizing its canonical decomposition into Seifert fibered spaces and hyperbolic manifolds.The algorithm relies on the extension of Matveev's theory of complexity to the caseof manifolds bounded by suitably marked tori, and on the notion of assembling of two such manifolds. We show that every manifold is an assembling of manifolds which cannot be further disassembled, and we prove that there are surprisingly few such manifolds up to complexity 9.Our most interesting experimental discovery is that there are 4 closed hyperbolic manifolds having complexity 9, and they are the 4 closed hyperbolic manifolds of least known volume. All other manifolds having complexity at most 9 are graph manifolds.