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Ivansic proved that there is a link $L$ of five tori in $S^4$ with hyperbolic complement. We describe $L$ explicitly and study its properties, and discover that $L$ is in many aspects similar to the Borromean rings in $S^3$. In particular the following hold: (1) Any two tori in $L$ are unlinked, but three are not; (2) The complement $M=S^4\setminus L$ is integral arithmetic hyperbolic; (3) The symmetry group of $L$ acts $k$-transitively on its components for all $k$; (4) The double branched covering over $L$ has geometry $\mathbb H^2 \times \mathbb H^2$; (5) The fundamental group of $M$ has a nice presentation via commutators; (6) The Alexander ideal is explicit; (7) Every class $x\in H^1(M, \mathbb Z)=\mathbb Z^5$ with $x_i\neq 0$ is represented by a perfect circle-valued Morse function; (8) By longitudinal Dehn surgery along $L$ we get a closed 4-manifold with fundamental group $\mathbb Z^5$; (9) The link $L$ can be put in perfect position. This leads also to the first descriptions of a cusped hyperbolic 4-manifold as a complement of tori in $\mathbb R \mathbb P^4$ and of some explicit Lagrangian tori in the product of two surfaces of genus two., Comment: 30 pages, 22 figures

Let $N$ be a complete finite-volume hyperbolic $n$-manifold. An efficient cycle for $N$ is the limit (in an appropriate measure space) of a sequence of fundamental cycles whose $\ell^1$-norm converges to the simplicial volume of $N$. Gromov and Thurston's smearing construction exhibits an explicit efficient cycle, and Jungreis and Kuessner proved that, in dimension $n\geq 3$, such cycle actually is the unique efficient cycle for a huge class of finite volume hyperbolic manifolds, including all the closed ones. In this paper we prove that, for $n\geq 3$, the class of finite-volume hyperbolic manifolds for which the uniqueness of the efficient cycle does not hold is exactly the commensurability class of the figure-8 knot complement (or, equivalently, of the Gieseking manifold)., Comment: 24 pages, 5 figures

We study how the spin structures on finite-volume hyperbolic n-manifolds restrict to cusps. When a cusp cross-section is a (n-1)-torus, there are essentially two possible behaviours: the spin structure is either bounding or Lie. We show that in every dimension n there are examples where at least one cusp is Lie, and in every dimension n <= 8 there are examples where all the cusps are bounding. By work of C. Bar, this implies that the spectrum of the Dirac operator is R in the first case, and discrete in the second. We therefore deduce that there are cusped hyperbolic manifolds whose spectrum of the Dirac operator is R in all dimensions, and whose spectrum is discrete in all dimensions n <= 8., Comment: 10 pages, 2 figures

We give examples of hyperbolic groups which contain subgroups that are of type $\mathscr{F}_{3}$ but not of type $\mathscr{F}_{4}$. These groups are obtained by Dehn filling starting from a non-uniform lattice in ${\rm PO}(8,1)$ which was previously studied by Italiano, Martelli and Migliorini., Comment: 24 pages, 1 figure

We exhibit some finite-volume cusped hyperbolic 5-manifolds that fiber over the circle. These include the smallest hyperbolic 5-manifold known, discovered by Ratcliffe and Tschantz. As a consequence, we build a finite type subgroup of a hyperbolic group that is not hyperbolic., Comment: 35 pages, 13 figures

We construct some cusped finite-volume hyperbolic $n$-manifolds $M_n$ that fiber algebraically in all the dimensions $5\leq n \leq 8$. That is, there is a surjective homomorphism $\pi_1(M_n) \to \mathbb Z$ 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 $\widetilde M_n$ whose fundamental group is finitely presented but not of finite type. These $n$-manifolds $\widetilde M_n$ have infinitely many cusps of maximal rank and hence infinite Betti number $b_{n-1}$. They cover the finite-volume manifold $M_n$. We obtain these examples by assigning some appropriate colours and states to a family of right-angled hyperbolic polytopes $P_5, \ldots, P_8$, and then applying some arguments of Jankiewicz, Norin, Wise and Bestvina, Brady. We exploit in an essential way the remarkable properties of the Gosset polytopes dual to $P_n$, and the algebra of integral octonions for the crucial dimensions $n=7,8$., Comment: 40 pages, 21 figures, final version

We exhibit some (compact and cusped) finite-volume hyperbolic four-manifolds M with perfect circle-valued Morse functions, that is circle-valued Morse functions $f\colon M \to S^1$ with only index 2 critical points. We construct in particular one example where every generic circle-valued function is homotopic to a perfect one. An immediate consequence is the existence of infinitely many finite-volume (compact and cusped) hyperbolic 4-manifolds $M$ having a handle decomposition with bounded numbers of 1- and 3-handles, so with bounded Betti numbers $b_1(M)$, $b_3(M)$ and rank of $\pi_1(M)$., Comment: 33 pages, 14 figures. The third version contains more examples and some proofs are expanded

We show that every plumbing of disc bundles over surfaces whose genera satisfy a simple inequality may be embedded as a convex submanifold in some closed hyperbolic four-manifold. In particular its interior has a geometrically finite hyperbolic structure that covers a closed hyperbolic four-manifold., Comment: 18 pages, 11 figures

We exhibit the first examples of compact orientable hyperbolic manifolds that do not have any spin structure. We show that such manifolds exist in all dimensions $n \geq 4$. The core of the argument is the construction of a compact orientable hyperbolic $4$-manifold $M$ that contains a surface $S$ of genus $3$ with self intersection $1$. The $4$-manifold $M$ has an odd intersection form and is hence not spin. It is built by carefully assembling some right angled $120$-cells along a pattern inspired by the minimum trisection of $\mathbb{C}\mathbb{P}^2$. The manifold $M$ is also the first example of a compact orientable hyperbolic $4$-manifold satisfying any of these conditions: 1) $H_2(M,\mathbb{Z})$ is not generated by geodesically immersed surfaces. 2) There is a covering $\tilde{M}$ that is a non-trivial bundle over a compact surface., Comment: 23 pages, 16 figures

We classify all the exceptional Dehn surgeries on the minimally twisted chain links with six and seven components., Comment: 34 pages

We study the set of all closed oriented smooth 4-manifolds experimentally, according to a suitable complexity defined using Turaev's shadows. This complexity roughly measures how complicated the 2-skeleton of the 4-manifold is. We characterise here all the closed oriented 4-manifolds that have complexity at most one. They are generated by a certain set of 20 blocks, that are some basic 4-manifolds with boundary consisting of copies of $S^2 \times S^1$, plus connected sums with some copies of $\mathbb{CP}^2$ with either orientation. All the manifolds generated by these blocks are doubles. Many of these are doubles of 2-handlebodies and are hence efficiently encoded using finite presentations of groups. In contrast to the complexity zero case, in complexity one there are also plenty of doubles that are not doubles of 2-handlebodies, like for instance $\mathbb{RP}^3 \times S^1$., Comment: 75 pages, 77 figures; introduction expanded in the second version

This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem., Comment: 486 pages, more than 200 colour figures. Second version of April 2022, with minor corrections

We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston's hyperbolic Dehn filling. We construct in particular an analytic path of complete, finite-volume cone four-manifolds $M_t$ that interpolates between two hyperbolic four-manifolds $M_0$ and $M_1$ with the same volume $\frac {8}3\pi^2$. The deformation looks like the familiar hyperbolic Dehn filling paths that occur in dimension three, where the cone angle of a core simple closed geodesic varies monotonically from $0$ to $2\pi$. Here, the singularity of $M_t$ is an immersed geodesic surface whose cone angles also vary monotonically from $0$ to $2\pi$. When a cone angle tends to $0$ a small core surface (a torus or Klein bottle) is drilled producing a new cusp. We show that various instances of hyperbolic Dehn fillings may arise, including one case where a degeneration occurs when the cone angles tend to $2\pi$, like in the famous figure-eight knot complement example. The construction makes an essential use of a family of four-dimensional deforming hyperbolic polytopes recently discovered by Kerckhoff and Storm., Comment: 60 pages, 23 figures. Final version

This is a short survey on finite-volume hyperbolic four-manifolds. We describe some general theorems and focus on the concrete examples that we found in the literature. The paper contains no new result., Comment: 20 pages, 5 figures

In this paper we raise the question whether every closed Riemannian manifold has a spine of minimal area, and we answer it affirmatively in the surface case. On constant curvature surfaces we introduce the spine systole, a continuous real function on moduli space that measures the minimal length of a spine in each surface. We show that the spine systole is a proper function and has its global minima precisely on the extremal surfaces (those containing the biggest possible discs). We also study minimal spines, which are critical points for the length functional. We completely classify minimal spines on flat tori, proving that the number of them is a proper function on moduli space. We also show that the number of minimal spines of uniformly bounded length is finite on hyperbolic surfaces., Comment: 24 pages, 12 figures

We prove that every complete finite-volume hyperbolic 3-manifold $M$ that is tessellated into (embedded) right-angled regular polyhedra (dodecahedra or ideal octahedra) embeds geodesically in a complete finite-volume connected orientable hyperbolic 4-manifold $W$, which is also tessellated into right-angled regular polytopes (120-cells and ideal 24-cells). If $M$ is connected, then Vol($W$) < $2^{49}$Vol($M$). This applies for instance to the Borromean link complement. As a consequence, the Borromean link complement bounds geometrically a hyperbolic 4-manifold., Comment: 11 pages, 6 figures. Mistake corrected: the decomposition needs to be "nice" in order to avoid self-adjacent (and hence uncolourable) facets

Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in $\mathbb{CP}^{n+1}$ decomposes into pairs of pants: a pair of pants is a real compact $2n$-manifold with cornered boundary obtained by removing an open regular neighborhood of $n+2$ generic hyperplanes from $\mathbb{CP}^n$. As is well-known, every compact surface of genus $g\geqslant 2$ decomposes into pairs of pants, and it is now natural to investigate this construction in dimension 4. Which smooth closed 4-manifolds decompose into pairs of pants? We address this problem here and construct many examples: we prove in particular that every finitely presented group is the fundamental group of a 4-manifold that decomposes into pairs of pants., Comment: 41 pages, 25 figures; exposition has been improved; the proof of Theorem 2 was incorrect, and it has been fixed. Accepted for publications in Algebr. Geom. Topol

Eisermann has shown that the Jones polynomial of a $n$-component ribbon link $L\subset S^3$ is divided by the Jones polynomial of the trivial $n$-component link. We improve this theorem by extending its range of application from links in $S^3$ to colored knotted trivalent graphs in $\#_g(S^2\times S^1)$, the connected sum of $g\geqslant 0$ copies of $S^2\times S^1$. We show in particular that if the Kauffman bracket of a knot in $\#_g(S^2\times S^1)$ has a pole in $q=i$ of order $n$, the ribbon genus of the knot is at least $\frac {n+1}2$. We construct some families of knots in $\#_g(S^2\times S^1)$ for which this lower bound is sharp and arbitrarily big. We prove these estimates using Turaev shadows., Comment: 38 pages, 18 figures

A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension. We construct here infinitely many explicit examples in dimension $n=3$ using right-angled dodecahedra and $120$-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every $k\geqslant 1$, we build an orientable compact closed $3$-manifold tessellated by $16k$ right-angled dodecahedra that bounds a $4$-manifold tessellated by $32k$ right-angled $120$-cells. A notable feature of this family is that the ratio between the volumes of the $4$-manifolds and their boundary components is constant and, in particular, bounded., Comment: 10 pages, 1 figure; ancillary Wolfram Mathematica notebook available at https://doi.org/10.7910/DVN/A6XZLC

We introduce a simple algorithm which transforms every four-dimensional cubulation into a cusped finite-volume hyperbolic four-manifold. Combinatorially distinct cubulations give rise to topologically distinct manifolds. Using this algorithm we construct the first examples of finite-volume hyperbolic four-manifolds with one cusp. More generally, we show that the number of $k$-cusped hyperbolic four-manifolds with volume smaller than V grows like $C^{V log V}$ for any fixed $k$. As a corollary, we deduce that the 3-torus bounds geometrically a hyperbolic manifold., Comment: 24 pages, 15 figures, typos corrected; Geom. and Funct. Anal., 2013

We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A|<1, and only densely defined when |A| = 1 and A is not a root of unity. When A is a root of unity distinct from 1, -1, i, -i the representations are finite-dimensional and isomorphic to the "Hom" version of the well-known TQFT quantum representations. The unitary representations in the interval [-1,0] interpolate analytically between two natural geometric unitary representations, the SU(2)-character variety representation studied by Goldman and the multicurve representation induced by the action of the mapping class group on multicurves. The finite-dimensional representations converge analytically to the infinite-dimensional ones. We recover Marche and Narimannejad's convergence theorem, and Andersen, Freedman, Walker and Wang's asymptotic faithfulness, that states that the image of a non-central mapping class is always non-trivial after some level r. When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level r in term of its dilatation., Comment: 41 pages, 13 figures

Let the complexity of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree we can promote it to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which call the "stable complexity" of M. We study here the relation between the stable complexity of M and Gromov's simplicial volume ||M||. It is immediate to show that ||M|| is smaller or equal than the stable complexity of M and it is natural to ask whether the two quantities coincide on aspherical manifolds with residually finite fundamental group. We show that this is not always the case: there is a constant C_n<1 such that ||M|| is smaller than C_n times the stable complexity for any hyperbolic manifold M of dimension at least 4. The question in dimension 3 is still open in general. We prove that the stable complexity equals ||M|| for any aspherical irreducible 3-manifold M whose JSJ decomposition consists of Seifert pieces and/or hyperbolic pieces commensurable with the figure-eight knot complement. The equality holds for all closed hyperbolic 3-manifolds if a particular three-dimensional version of the Ehrenpreis conjecture is true., Comment: 35 pages, 5 figures, minor corrections and few arguments added following referees' suggestions

We consider in this paper the minimally twisted chain link with 5 components in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn fillings on its exterior M5. The 3-manifold M5 is a nicely symmetric hyperbolic one, filling which one gets a wealth of hyperbolic 3-manifolds having 4 or fewer (including 0) cusps. In view of Thurston's hyperbolic Dehn filling theorem it is then natural to face the problem of classifying all the exceptional fillings on M5, namely those yielding non-hyperbolic 3-manifolds. Here we completely solve this problem, also showing that, thanks to the symmetries of M5 and of some hyperbolic manifolds resulting from fillings of M5, the set of exceptional fillings on M5 is described by a very small amount of information., Comment: 46 pages, 10 figures. This new version uses the "hikmot" libraries to verify the numeric computations made by SnapPy

We exhibit a finite set of local moves that connect any two surgery presentations of the same 3-manifold via framed links in the three-sphere. The moves are handle-slides and blow-downs/ups of a particular simple kind., Comment: 5 pages, 9 figures, minor changes following referee's suggestions

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 x S^1, plus some complex projective spaces. We deduce a classification of all 4-manifolds with finite fundamental group and shadow-complexity zero., Comment: 63 pages, 75 figures

We extend Matveev's complexity of 3-manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant). Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspherical manifolds (in particular, on manifolds with non-positive sectional curvature). There are finitely many closed hyperbolic manifolds of any given complexity. On the other hand, there are many closed 4-manifolds of complexity zero (manifolds without 3-handles, doubles of 2-handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary fundamental group)., Comment: 42 pages, 24 figures. In the new version, we have expanded the introduction

In this paper we enumerate and classify the ``simplest'' pairs (M,G) where M is a closed orientable 3-manifold and G is a trivalent graph embedded in M. To enumerate the pairs we use a variation of Matveev's definition of complexity for 3-manifolds, and we consider only (0,1,2)-irreducible pairs, namely pairs (M,G) such that any 2-sphere in M intersecting G transversely in at most 2 points bounds a ball in M either disjoint from G or intersecting G in an unknotted arc. To classify the pairs our main tools are geometric invariants defined using hyperbolic geometry. In most cases, the graph complement admits a unique hyperbolic structure with parabolic meridians; this structure was computed and studied using Heard's program Orb and Goodman's program Snap. We determine all (0,1,2)-irreducible pairs up to complexity 5, allowing disconnected graphs but forbidding components without vertices in complexity 5. The result is a list of 129 pairs, of which 123 are hyperbolic with parabolic meridians. For these pairs we give detailed information on hyperbolic invariants including volumes, symmetry groups and arithmetic invariants. Pictures of all hyperbolic graphs up to complexity 4 are provided. We also include a partial analysis of knots and links. The theoretical framework underlying the paper is twofold, being based on Matveev's theory of spines and on Thurston's idea (later developed by several authors) of constructing hyperbolic structures via triangulations. Many of our results were obtained (or suggested) by computer investigations., Comment: 35 pages, 22 figures, 10 tables

Following Matveev, a k-normal surface in a triangulated 3-manifold is a generalization of both normal and (octagonal) almost normal surfaces. Using spines, complexity, and Turaev-Viro invariants of 3-manifolds, we prove the following results: 1) a minimal triangulation of a closed irreducible or a bounded hyperbolic 3-manifold contains no non-trivial k-normal sphere; 2) every triangulation of a closed manifold with at least 2 tetrahedra contains some non-trivial normal surface; 3) every manifold with boundary has only finitely many triangulations without non-trivial normal surfaces. Here, triangulations of bounded manifolds are actually ideal triangulations. We also calculate the number of normal surfaces of nonnegative Euler characteristics which are contained in the conjecturally minimal triangulations of all lens spaces., Comment: 12 pages, 7 figures

We prove that for every countable group G there exists a hyperbolic 3-manifold M such that the isometry group of M, the mapping class group of M, and the outer automorphism group of the fundamental group of M are isomorphic to G., Comment: 15 pages, 6 figures

We show that only finitely many links in a closed 3-manifold share the same complement, up to twists along discs and annuli. Using the same techniques, we prove that by adding 2-handles on the same link we get only finitely many smooth cobordisms between two given closed 3-manifolds. As a consequence, there are finitely many smooth closed 4-manifolds constructed from some Kirby diagram with bounded number of crossings, discs, and strands, or from some Turaev special shadow with bounded number of vertices. (These are the 4-dimensional analogues of Heegaard diagrams and special spines for 3-manifolds.) We therefore get two filtrations on the set of all smooth closed 4-manifolds with finite sets. The two filtrations are equivalent after linear rescalings, and their cardinality grows at least as n^{c*n}., Comment: 23 pages, 9 figures. Final version

We give a summary of known results on Matveev's complexity of compact 3-manifolds. The only relevant new result is the classification of all closed orientable irreducible 3-manifolds of complexity 10., Comment: 26 pages, 7 figures, minor corrections

We establish a bijective correspondence between the set T(n) of 3-dimensional triangulations with n tetrahedra and a certain class H(n) of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus n+1. We show that the manifolds in H(n) are hyperbolic (with geodesic boundary, and cusps corresponding to the loops), have least possible volume, and simplest boundary loops. Mirroring the elements of H(n) in their geodesic boundary we obtain a class D(n) of cusped hyperbolic manifolds, previously considered by D. Thurston and the first named author. We show that also D(n) corresponds bijectively to T(n), and we study some Dehn fillings of the manifolds in D(n). As consequences of our constructions, we also show that: - A triangulation of a 3-manifold is uniquely determined up to isotopy by its 1-skeleton; - If a 3-manifold M has an ideal triangulation with edges of valence at least 6, then M is hyperbolic and the edges are homotopically non-trivial, whence homotopic to geodesics; - Every finite group G is the isometry group of a closed hyperbolic 3-manifold with volume less than a constant times |G|^9., Comment: Revised version, accepted for publication in Commentarii Mathematici Helvetici. 34 pages, 22 figures

We classify all closed non-orientable P2-irreducible 3-manifolds with complexity up to 7, fixing two mistakes in our previous complexity-up-to-6 classification. We show that there is no such manifold with complexity less than 6, five with complexity 6 (the four flat ones and the filling of the Gieseking manifold, which is of type Sol, and three with complexity 7 (one manifold of type Sol, and the two manifolds of type H2xR with smallest base orbifolds)., Comment: 20 pages, 10 figures, 2 tables

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 M6_1^3 (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., Comment: 62 pages, 9 figures, this is the second part of (a revised version of) the former math.GT/0204228

We define for each g>=2 and k>=0 a set M_{g,k} of orientable hyperbolic 3-manifolds with $k$ toric cusps and a connected totally geodesic boundary of genus g. Manifolds in M_{g,k} have Matveev complexity g+k and Heegaard genus g+1, and their homology, volume, and Turaev-Viro invariants depend only on g and k. In addition, they do not contain closed essential surfaces. The cardinality of M_{g,k} for a fixed k has growth type g^g. We completely describe the non-hyperbolic Dehn fillings of each M in M_{g,k}, showing that, on any cusp of any hyperbolic manifold obtained by partially filling M, there are precisely 6 non-hyperbolic Dehn fillings: three contain essential discs, and the other three contain essential annuli. This gives an infinite class of large hyperbolic manifolds (in the sense of Wu) with boundary-reducible and annular Dehn fillings having distance 2, and allows us to prove that the corresponding upper bound found by Wu is sharp. If M has one cusp only, the three boundary-reducible fillings are handlebodies., Comment: 28 pages, 16 figures

We classify the orientable finite-volume hyperbolic 3-manifolds having non-empty 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., Comment: 21 pages, 6 figures

We classify all closed non-orientable P2-irreducible 3-manifolds having complexity up to 6 and we describe some having complexity 7. We show in particular that there is no such manifold with complexity less than 6, and that those having complexity 6 are precisely the 4 flat non-orientable ones and the filling of the Gieseking manifold, which is of type Sol. The manifolds having complexity 7 we describe are Seifert manifolds of type H2 x S1 and a manifold of type Sol., Comment: 27 pages, 12 figures. Two mistakes contained in the previous version are fixed: there is a Sol manifold with complexity 6, and the examples with complexty 7 are Sol and H2xS1 (see the abstract)

Using the theory of hyperbolic manifolds with totally geodesic boundary, we provide for every integer n greater than 1 a class of such manifolds all having Matveev complexity equal to n and Heegaard genus equal to n+1. All the elements of this class have a single boundary component of genus n, and the numbers of distinct members of the class grows at least exponentially with n., Comment: 15 pages, 7 figures

We classify all the non-hyperbolic Dehn fillings of the complement of the chain-link with 3 components, conjectured to be the smallest hyperbolic 3-manifold with 3 cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many 1-cusped and 2-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 the 3-sphere 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 2-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., Comment: 56 pages, 10 figures, 16 tables. Some consequences of the classification added

Let M be a (possibly non-orientable) compact 3-manifold with (possibly empty) boundary consisting of tori and Klein bottles. Let $X\subset\partial M$ be a trivalent graph such that $\partial M\setminus X$ is a union of one disc for each component of $\partial M$. Building on previous work of Matveev, we define for the pair (M,X) a complexity c(M,X) and show that, when M is closed, irreducible and P^2-irreducible, $c(M,\emptyset)$ is the minimal number of tetrahedra in a triangulation of M. Moreover c is additive under connected sum, and, given any n>=0, there are only finitely many irreducible and P^2-irreducible closed manifolds having complexity up to n. We prove that every irreducible and P^2-irreducible pair (M,X) has a finite splitting along tori and Klein bottles into pairs having the same properties, and complexity is additive on this splitting. As opposed to the JSJ decomposition, our splitting is not canonical, but it involves much easier blocks than all Seifert and simple manifolds. In particular, most Seifert and hyperbolic manifolds appear to have non-trivial splitting. In addition, a given set of blocks can be combined to give only a finite number of pairs (M,X). Our splitting theorem provides the theoretical background for an algorithm which classifies 3-manifolds of any given complexity. This algorithm has been already implemented and proved effective in the orientable case for complexity up to 9., Comment: 32 pages, 16 figures

We describe an algorithm which has enabled us to give a complete list, without repetitions, of all closed oriented irreducible 3-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 fibred spaces and hyperbolic manifolds. The algorithm relies on the extension of Matveev's theory of complexity to the case of 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 deassembled, and we prove that there are surprisingly few such manifolds up to complexity 9. Our most interesting experimental discovery is the following: there are 4 hyperbolic manifolds having complexity 9, and they turn out to be precisely the 4 closed hyperbolic manifolds of least known volume. All other manifolds having complexity at most 9 are graph manifolds., Comment: 49 pages, 32 figures. This is a revised version of math.GT/0005104, accepted for publication in Experimental Mathematics. The main improvement is an intrinsic definition of "brick", in terms of manifolds rather than in terms of spines, and an extension of Matveev's complexity to manifolds bounded by suitably marked tori