We classify all the non-hyperbolic Dehn fillings of the complement of the chain link with three components, conjectured to be the smallest hyperbolic 3-manifold with three cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many one-cusped and two-cusped hyperbolic manifolds, including most of those with smallest known volume.Among other consequences of this classification, we mention the following:• for every integer n, we can prove that there are infinitely many hyperbolic knots in S 3 having exceptional surgeries {n, n + 1, n + 2, n + 3}, with n + 1, n + 2 giving small Seifert manifolds and n, n + 3 giving toroidal manifolds.• we exhibit a two-cusped hyperbolic manifold that contains a pair of inequivalent knots having homeomorphic complements.• we exhibit a chiral 3-manifold containing a pair of inequivalent hyperbolic knots with orientation-preservingly homeomorphic complements.• we give explicit lower bounds for the maximal distance between small Seifert fillings and any other kind of exceptional filling. IntroductionWe study in this paper the Dehn fillings of the complement N of the chain link with three components in S 3 , shown in figure 1. The hyperbolic structure of N was first constructed by Thurston in his notes [28], and it was also noted there that the volume of N is particularly small. The relevance of N to three-dimensional topology comes from the fact that by filling N , one gets most of the hyperbolic manifolds known and most of the interesting non-hyperbolic fillings of cusped hyperbolic manifolds. For these reasons N was called the "magic manifold" by Gordon and Wu [14,17]. It appears as M 6 3 1 in [6] and it is the hyperbolic manifold with three cusps of smallest known volume and of smallest complexity [1]. (We refer here to the complexity defined by Matveev in [23], and we mean that N has an ideal triangulation with six tetrahedra, while all other hyperbolic manifolds 969

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We prove that a closed 4-manifold has shadow-complexity zero if and only if it is a kind of 4-dimensional graph manifold, which decomposes into some particular blocks along embedded copies of S 2 × S 1 , plus some complex projective spaces. We deduce a classification of all 4-manifolds with finite fundamental group and shadow-complexity zero.

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, we describe their canonical Kojima decomposition, and we discuss manifolds having cusps.The 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). And there is a single cusped manifold, that we can show to be a knot complement in a genus-2 handlebody. Concerning manifolds built from four tetrahedra, we show that there are 5033 different ones, with 262 different volumes. The Kojima decomposition consists either of tetrahedra (as many as eight of them in some cases), or 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.MSC (2000): 57M50 (primary), 57M20, 57M27 (secondary). This paper is devoted to the class of all orientable finite-volume hyperbolic 3manifolds having non-empty compact totally geodesic boundary and admitting a minimal ideal triangulation with either three or four but no fewer tetrahedra. We describe the theoretical background and experimental results of a computer program that has enabled us to classify all such manifolds. (The case of manifolds obtained from two tetrahedra was previously dealt with in [8]). We also provide an overall discussion of the most important features of all these manifolds, namely of:• their volumes;• the shape of their canonical Kojima decomposition; Preliminaries and statementsWe consider in this paper the class H of orientable 3-manifolds M having compact non-empty boundary ∂M and admitting a complete finite-volume hyperbolic metric with respect to which ∂M is totally geodesic. It is a well-known fact [9] that such an M is the union of a compact portion and some cusps based on tori, so it has a natural compactification obtained by adding some tori. The elements of H are regarded up to homeomorphism, or equivalently isometry (by Mostow's rigidity). Candidate hyperbolic manifolds Let us now introduce the class H of 3-manifolds M such that:• M is orientable, compact, boundary-irreducible and acylindrical;• ∂M consists of some tori (possibly none of them) and at least one surface of negative Euler characteristic.The basic theory of hyperbolic manifolds implies that, up to identifying a manifold with its natural compactification, the inclusion H ⊂ H holds. We note that, by Thurston's hyperbolization, an element of H actually lies in H if and only if it is atoroidal. However we do not require atoroidality in the definition of H, for a reason that will be mentioned later in this section and explained in detail in...

We construct some cusped finite-volume hyperbolic n-manifolds Mn that fiber algebraically in all the dimensions 5 ≤ n ≤ 8. That is, there is a surjective homomorphism π 1 (Mn) → Z with finitely generated kernel.The kernel is also finitely presented in the dimensions n = 7, 8, and this leads to the first examples of hyperbolic n-manifolds Mn whose fundamental group is finitely presented but not of finite type. These n-manifolds Mn have infinitely many cusps of maximal rank and hence infinite Betti number b n−1 . They cover the finite-volume manifold Mn.We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes P 5 , . . . , P 8 , and then applying some arguments of Jankiewicz -Norin -Wise [15] and . We exploit in an essential way the remarkable properties of the Gosset polytopes dual to Pn, and the algebra of integral octonions for the crucial dimensions n = 7, 8.